%PDF-1.3 % 2 0 obj << /Length 7020 >> stream BT /F1 1 Tf 7.9701 0 0 7.9701 224.0259 729.066 Tm 0 g /GS1 gs 0 Tc 0 Tw [(Annals)-354.1(of)-354.2(Mathematics)]TJ /F2 1 Tf 10.9091 0 0 10.9091 309.9637 729.066 Tm (,)Tj /F3 1 Tf 7.9701 0 0 7.9701 316.6306 729.066 Tm (165)Tj /F1 1 Tf 2.2938 0 TD [(\(2007\),)-354(143)]TJ /F4 1 Tf 17.2154 0 0 17.2154 199.7509 647.6025 Tm [(An)-375(uncertain)31.3(t)31.3(y)-375(principle)]TJ 0.0771 -1.1574 TD [(for)-375(arithmetic)-375(sequences)]TJ /F2 1 Tf 9.9626 0 0 9.9626 188.2202 595.5377 Tm (By)Tj /F5 1 Tf 1.5695 0 TD [(Andrew)-377.7(Granville)]TJ /F2 1 Tf 9.9385 0 TD (and)Tj /F5 1 Tf 1.9445 0 TD [(K.)-377.7(Sound)22.2(ararajan)]TJ /F2 1 Tf 9.3113 0 TD (*)Tj /F6 1 Tf 10.9091 0 0 10.9091 280.1036 560.8677 Tm (Abstract)Tj /F2 1 Tf -12.7771 -2.3653 TD [(Analytic)-330.7(n)27.8(u)0(m)27.8(b)-27.7(er)-330.8(theorists)-330.7(usually)-330.7(seek)-330.8(to)-330.7(sho)27.7(w)-330.7(that)-330.7(sequences)-330.7(whic)27.8(h)-330.8(ap-)]TJ -2.0091 -1.2694 TD [(p)-27.8(ear)-334.8(naturally)-334.8(in)-334.9(arithmetic)-334.8(are)-334.9(w)27.8(ell-distributed)-334.9(i)0(n)-335(some)-334.9(appropriate)-334.7(sense.)]TJ T* [(In)-291(v)55.5(arious)-290.9(discrepancy)-290.8(problems,)-299.4(com)27.8(binatorics)-290.8(researc)27.8(hers)-290.9(ha)27.8(v)27.8(e)-291(analyzed)-291(lim-)]TJ T* [(itations)-366.1(to)-366.2(equi-distribution,)-374.2(as)-366.1(ha)27.8(v)27.8(e)-366.2(F)83.4(ourier)-366.1(analysts)-366(when)-366.1(w)27.8(orking)-366.1(with)-366.1(the)]TJ T* [(uncertain)27.9(t)27.8(y)-323.5(principle.)-441.1(In)-323.6(this)-323.5(article)-323.5(w)27.8(e)-323.6(nd)-323.5(that)-323.5(these)-323.5(ideas)-323.5(ha)27.8(v)27.8(e)-323.6(a)-323.6(natural)]TJ T* [(setting)-311.3(in)-311.4(the)-311.4(analysis)-311.3(of)-311.4(distributions)-311.3(of)-311.4(sequences)-311.2(in)-311.4(analytic)-311.3(n)27.8(u)0(m)27.8(b)-27.7(er)-311.5(theory)83.5(,)]TJ T* [(form)27.9(ulating)-333.3(a)-333.3(general)-333.2(principle,)-333.2(and)-333.2(giving)-333.3(sev)27.8(eral)-333.3(examples.)]TJ /F6 1 Tf 13.0815 -2.9132 TD [(1.)-574.9(In)31.9(tro)-31.9(duction)]TJ /F2 1 Tf -11.0723 -2.3653 TD [(In)-357.7(this)-357.8(pap)-27.7(er)-357.8(w)27.8(e)-357.7(in)27.8(v)27.8(estigate)-357.7(the)-357.7(limitations)-357.7(to)-357.8(the)-357.7(equidistribution)-357.5(of)-357.8(in-)]TJ -2.0091 -1.2694 TD [(teresting)-255.9(arithmetic)-255.8(sequences)-255.9(i)0(n)-256(arithmetic)-255.8(progressions)-255.8(and)-255.9(short)-255.9(in)27.8(terv)55.6(als.)]TJ T* [(Our)-300.3(discussions)-300.4(are)-300.4(motiv)55.6(ated)-300.4(b)27.8(y)-300.4(a)-300.4(general)-300.4(result)-300.4(of)-300.4(K.)-300.4(F.)-300.4(Roth)-300.4([15])-300.4(on)-300.4(irregu-)]TJ T* [(larities)-341(of)-341(distribution,)-342.9(and)-341(a)-341.1(particular)-340.9(result)-341(of)-341.1(H.)-341.1(Maier)-341([11])-341(whic)27.8(h)-341.1(imp)-27.7(oses)]TJ T* [(restrictions)-333.2(on)-333.3(the)-333.3(equidistribution)-333.2(of)-333.3(primes.)]TJ 2.0091 -1.2694 TD (If)Tj /F7 1 Tf 0.9263 0 TD (A)Tj /F2 1 Tf 1.0581 0 TD [(is)-259.6(a)-259.6(subset)-259.6(of)-259.6(the)-259.6(in)27.8(tegers)-259.6(in)-259.6([1)]TJ /F8 1 Tf 12.898 0 TD 0.1667 Tc (,x)Tj /F2 1 Tf 1.016 0 TD 0 Tc [(])-259.7(with)]TJ /F7 1 Tf 2.7415 0 TD (|A|)Tj /F2 1 Tf 1.6319 0 TD (=)Tj /F8 1 Tf 1.0555 0 TD (x)Tj /F2 1 Tf 1.3481 0 TD [(then,)-274.3(as)-259.6(Roth)-259.6(pro)27.8(v)27.8(ed,)]TJ -24.6845 -1.2694 TD [(there)-285.6(exists)]TJ /F8 1 Tf 5.2242 0 TD (N)Tj /F7 1 Tf 1.1903 0 TD ()Tj /F8 1 Tf 1.0555 0 TD (x)Tj /F2 1 Tf 0.8573 0 TD [(and)-285.7(an)-285.7(arithmetic)-285.6(progression)]TJ /F8 1 Tf 13.2179 0 TD (a)Tj /F2 1 Tf 0.9731 0 TD [(\(mo)-27.7(d)]TJ /F8 1 Tf 2.6388 0 TD (q)Tj /F2 1 Tf 0.4824 0 TD [(\))-285.7(with)]TJ /F8 1 Tf 2.9048 0 TD (q)Tj /F7 1 Tf 0.7601 0 TD ()Tj 1.0555 0.7203 TD ()Tj ET 459.09 319.606 6.235 -0.436 re f BT /F8 1 Tf 10.9091 0 0 10.9091 459.0884 311.3036 Tm (x)Tj /F2 1 Tf 0.8573 0 TD [(suc)27.8(h)]TJ -32.0504 -1.2694 TD (that)Tj /F9 1 Tf 7.9225 -1.3405 TD ()Tj 0 -0.6 TD ()Tj T* ()Tj 1.9519 1 TD ()Tj /F10 1 Tf 4.9813 0 0 4.9813 216.276 258.8862 Tm (n)Tj /F11 1 Tf 0.8808 0 TD (A)Tj /F10 1 Tf 2.058 0 TD 0.7299 Tc (,n)Tj /F11 1 Tf 2.069 0 TD 0 Tc ()Tj /F10 1 Tf 1.0834 0 TD (N)Tj -6.9744 -1.3594 TD (n)Tj /F11 1 Tf 0.8808 0 TD ()Tj /F10 1 Tf 1.0834 0 TD (a)Tj /F12 1 Tf 2.2466 0 TD -0.0002 Tc [(\(mo)-35.2(d)]TJ /F10 1 Tf 3.595 0 TD 0 Tc (q)Tj /F12 1 Tf 0.6948 0 TD (\))Tj /F2 1 Tf 10.9091 0 0 10.9091 259.9323 270.2874 Tm (1)Tj /F7 1 Tf 0.7223 0 TD ()Tj /F2 1 Tf 1.1096 0.6765 TD (1)Tj ET 279.93 273.286 5.455 -0.436 re f BT /F8 1 Tf 10.9091 0 0 10.9091 280.0129 262.8044 Tm (q)Tj /F9 1 Tf 0.8737 1.6359 TD ()Tj /F10 1 Tf 4.9813 0 0 4.9813 290.1031 259.0655 Tm (n)Tj /F11 1 Tf 0.8808 0 TD (A)Tj /F10 1 Tf -0.9858 -1.3594 TD (n)Tj /F11 1 Tf 0.8808 0 TD ()Tj /F10 1 Tf 1.0832 0 TD (N)Tj /F2 1 Tf 10.9091 0 0 10.9091 308.279 270.2874 Tm (1)Tj /F9 1 Tf 0.8333 1.1499 TD ()Tj 0 -0.6 TD ()Tj T* ()Tj /F7 1 Tf 0.6111 0.0501 TD ()Tj /F9 1 Tf 1.2777 0.9038 TD ()Tj ET 348.93 280.606 38.553 -0.436 re f BT /F8 1 Tf 10.9091 0 0 10.9091 348.8847 270.2874 Tm ()Tj /F2 1 Tf 0.5171 0 TD (\(1)Tj /F7 1 Tf 1.1111 0 TD ()Tj /F8 1 Tf 1 0 TD ()Tj /F2 1 Tf 0.5171 0 TD (\))Tj /F8 1 Tf 0.3889 0 TD (x)Tj /F12 1 Tf 4.9813 0 0 4.9813 394.8683 278.0504 Tm (1)Tj ET 394.89 276.943 3.39 -0.373 re f BT 4.9813 0 0 4.9813 394.8683 271.6662 Tm (4)Tj /F8 1 Tf 10.9091 0 0 10.9091 399.9522 270.2874 Tm (.)Tj /F2 1 Tf -25.7723 -4.5173 TD [(In)-418.5(other)-418.4(w)27.7(ords,)-439.7(k)27.8(eeping)]TJ 10.9841 0 TD -0.0278 Tc (away)Tj 2.5851 0 TD 0 Tc [(from)-418.4(sets)-418.5(of)-418.4(densit)27.8(y)-418.4(0)-418.5(or)-418.5(1,)-439.8(there)-418.4(m)27.8(ust)-418.4(b)-27.8(e)-418.5(an)]TJ -13.5692 -1.2694 TD [(arithmetic)-269.1(progression)-269(in)-269.1(whic)27.8(h)-269.2(the)-269.2(n)27.8(u)0(m)27.8(b)-27.7(er)-269.2(of)-269.1(elemen)27.8(ts)-269.2(of)]TJ /F7 1 Tf 25.2558 0 TD (A)Tj /F2 1 Tf 1.0677 0 TD [(is)-269.1(a)-269.1(little)-269.1(dieren)27.8(t)]TJ -26.3235 -1.2694 TD [(from)-304.5(the)-304.5(a)27.8(v)27.8(erage.)-434.7(F)83.4(ollo)27.7(wing)-304.4(w)27.7(ork)-304.5(of)-304.5(A.)-304.5(Sark)27.8(ozy)-304.5(and)-304.5(J.)-304.6(Bec)27.9(k,)-310.2(J.)-304.6(Matousek)-304.3(and)]TJ T* [(J.)-328(Sp)-27.7(encer)-327.9([12])-327.9(sho)27.7(w)27.8(e)0(d)-327.9(that)-327.8(Roths)-327.9(theorem)-327.9(is)-327.9(b)-27.8(est)-327.9(p)-27.7(ossible,)-328.9(in)-327.9(that)-327.9(there)-327.8(is)-327.9(a)]TJ ET 118.77 159.849 148.242 -0.399 re f BT /F13 1 Tf 8.9664 0 0 8.9664 135.3873 149.1407 Tm [(*Le)-403.5(premier)-403.5(auteur)-403.5(est)-403.5(partiellemen)28.7(t)-403.6(souten)28.6(u)-403.6(par)-403.5(une)-403.5(b)-28.6(ourse)-403.5(du)-403.6(Conseil)-403.4(de)-403.6(rec)28.5(herc)28.7(hes)]TJ -1.85 -1.2222 TD [(en)-389.2(sciences)-389.1(naturelles)-389.1(et)-389.2(en)-389.2(g)28.4()]TJ 13.2164 0 TD [(enie)-389.1(du)-389.1(Canada.)-596.4(The)-389.2(second)-389(author)-389.2(is)-389.1(partially)-389.1(supp)-28.5(orted)-389.1(b)28.5(y)]TJ -13.2164 -1.2222 TD [(the)-342.6(National)-342.5(Science)-342.5(F)85.7(oundation.)]TJ ET endstream endobj 3 0 obj << /ProcSet [/PDF /Text ] /Font << /F1 4 0 R /F2 5 0 R /F3 6 0 R /F4 7 0 R /F5 8 0 R /F6 9 0 R /F7 10 0 R /F8 11 0 R /F9 12 0 R /F10 13 0 R /F11 14 0 R /F12 15 0 R /F13 16 0 R >> /ExtGState << /GS1 17 0 R >> >> endobj 20 0 obj << /Length 12939 >> stream BT /F2 1 Tf 10.9091 0 0 10.9091 118.7995 726.9739 Tm 0 g /GS1 gs 0 Tc 0 Tw (2)Tj /F5 1 Tf 7.9701 0 0 7.9701 199.3913 726.9739 Tm [(ANDREW)-377.5(GRANVILLE)-377.5(AND)-377.6(K)-0.1(.)-377.7(SOUND)29.3(ARARAJAN)]TJ /F2 1 Tf 10.9091 0 0 10.9091 118.7995 701.071 Tm (set)Tj /F7 1 Tf 1.5611 0 TD (A)Tj /F2 1 Tf 1.1318 0 TD [(con)27.8(taining)]TJ /F7 1 Tf 4.8612 0 TD ()Tj /F8 1 Tf 1.0555 0 TD (x/)Tj /F2 1 Tf 1.0715 0 TD [(2)-333.3(i)0(n)27.8(tegers)-333.3(up)-333.3(to)]TJ /F8 1 Tf 7.2028 0 TD (x)Tj /F2 1 Tf 0.5715 0 TD [(,)-333.4(for)-333.3(whic)27.8(h)]TJ /F7 1 Tf -15.8319 -1.6888 TD (|)Tj /F2 1 Tf 0.2777 0 TD (#)Tj /F7 1 Tf 0.8333 0 TD ({)Tj /F8 1 Tf 0.5 0 TD (n)Tj /F7 1 Tf 0.878 0 TD 0.2778 Tc (A)Tj /F2 1 Tf 2.0207 0 TD 0 Tc (:)Tj /F8 1 Tf 0.8889 0 TD (n)Tj /F7 1 Tf 0.878 0 TD ()Tj /F8 1 Tf 1.0556 0 TD 0.0534 Tc [(N,)-446.6(n)]TJ /F7 1 Tf 2.5127 0 TD 0 Tc ()Tj /F8 1 Tf 1.0555 0 TD (a)Tj /F2 1 Tf 1.5286 0 TD [(\(mo)-27.8(d)]TJ /F8 1 Tf 2.6389 0 TD (q)Tj /F2 1 Tf 0.4823 0 TD (\))Tj /F7 1 Tf 0.3889 0 TD 0.2223 Tc (})Tj /F2 1 Tf 1.7223 0 TD 0 Tc (#)Tj /F7 1 Tf 0.8333 0 TD ({)Tj /F8 1 Tf 0.5 0 TD (n)Tj /F7 1 Tf 0.878 0 TD 0.2778 Tc (A)Tj /F2 1 Tf 2.0206 0 TD 0 Tc (:)Tj /F8 1 Tf 0.889 0 TD (n)Tj /F7 1 Tf 0.878 0 TD ()Tj /F8 1 Tf 1.0555 0 TD (N)Tj /F7 1 Tf 0.9125 0 TD (})Tj /F8 1 Tf 0.5 0 TD (/q)Tj /F7 1 Tf 0.9823 0 TD 0.2777 Tc (|)Tj /F8 1 Tf 1.8333 0 TD 0 Tc (x)Tj /F1 1 Tf 7.9701 0 0 7.9701 458.4977 687.1522 Tm (1)Tj /F14 1 Tf 0.5312 0 TD (/)Tj /F1 1 Tf 0.5312 0 TD (4)Tj /F2 1 Tf 10.9091 0 0 10.9091 118.7995 664.2242 Tm [(for)-333.3(all)]TJ /F8 1 Tf 2.9195 0 TD (q)Tj /F2 1 Tf 0.8156 0 TD (and)Tj /F8 1 Tf 1.9445 0 TD (a)Tj /F2 1 Tf 0.8619 0 TD (with)Tj /F8 1 Tf 2.2778 0 TD (N)Tj /F7 1 Tf 1.1902 0 TD ()Tj /F8 1 Tf 1.0555 0 TD (x)Tj /F2 1 Tf 0.5716 0 TD (.)Tj -9.6275 -1.2694 TD [(Roths)-447.1(result)-447(concerns)-447.1(arbitrary)-446.9(sequences)-447(of)-447(in)27.8(tegers,)-475.5(as)-447.1(considered)-446.9(in)]TJ -2.0091 -1.2694 TD [(com)27.8(binatorial)-355.2(n)27.8(u)0(m)27.7(b)-27.7(er)-355.3(theory)-355.3(and)-355.3(harmonic)-355.3(analysis.)-510.3(W)83.3(e)-355.4(are)-355.3(more)-355.3(in)27.8(terested)]TJ T* [(here)-420.6(in)-420.7(sets)-420.7(of)-420.7(in)27.9(tegers)-420.7(that)-420.7(arise)-420.7(in)-420.6(arithmetic,)-442.4(suc)27.8(h)-420.7(a)0(s)-420.7(the)-420.7(primes.)-706.5(In)-420.8([11])]TJ T* [(H.)-301.7(Maier)-301.7(dev)27.8(elop)-27.8(ed)-301.7(an)-301.7(ingenious)-301.6(metho)-27.8(d)-301.7(t)0(o)-301.8(sho)27.8(w)-301.7(that)-301.7(for)-301.7(an)27.8(y)]TJ /F8 1 Tf 27.2247 0 TD (A)Tj /F7 1 Tf 1.0279 0 TD ()Tj /F2 1 Tf 1.0555 0 TD [(1)-301.7(there)-301.7(are)]TJ -29.308 -1.2694 TD [(arbitrarily)-298(large)]TJ /F8 1 Tf 7.1908 0 TD (x)Tj /F2 1 Tf 0.8697 0 TD [(suc)27.8(h)-298.1(that)-298.1(the)-298.1(in)27.8(terv)55.6(al)-298.1(\()]TJ /F8 1 Tf 10.0065 0 TD [(x,)-166.7(x)]TJ /F2 1 Tf 1.7394 0 TD [(+)-151.9(\(log)]TJ /F8 1 Tf 2.7769 0 TD (x)Tj /F2 1 Tf 0.5715 0 TD (\))Tj /F14 1 Tf 7.9701 0 0 7.9701 375.6394 598.943 Tm (A)Tj /F2 1 Tf 10.9091 0 0 10.9091 382.4797 594.9838 Tm [(\))-298.2(con)27.9(tains)-298.2(signican)27.8(tly)]TJ -24.1707 -1.2694 TD [(more)-297.3(primes)-297.2(than)-297.3(usual)-297.2(\(that)-297.3(is,)]TJ /F7 1 Tf 14.3133 0 TD ()Tj /F2 1 Tf 1.0555 0 TD [(\(1)-150.2(+)]TJ /F8 1 Tf 1.967 0 TD ()Tj /F14 1 Tf 7.9701 0 0 7.9701 312.7661 579.4471 Tm (A)Tj /F2 1 Tf 10.9091 0 0 10.9091 319.6075 581.1357 Tm (\)\(log)Tj /F8 1 Tf 2.2362 0 TD (x)Tj /F2 1 Tf 0.5715 0 TD (\))Tj /F14 1 Tf 7.9701 0 0 7.9701 354.4787 585.0949 Tm (A)Tj /F15 1 Tf 0.7959 0 TD ()Tj /F1 1 Tf 0.8264 0 TD (1)Tj /F2 1 Tf 10.9091 0 0 10.9091 375.3843 581.1357 Tm [(primes)-297.2(for)-297.3(some)]TJ /F8 1 Tf 7.1585 0 TD ()Tj /F14 1 Tf 7.9701 0 0 7.9701 458.3263 579.4471 Tm (A)Tj /F8 1 Tf 10.9091 0 0 10.9091 468.1973 581.1357 Tm (>)Tj /F2 1 Tf 1.0556 0 TD (0\))Tj -33.0837 -1.2694 TD [(and)-375.3(also)-375.2(in)27.8(terv)55.6(als)-375.3(\()]TJ /F8 1 Tf 8.4732 0 TD [(x,)-166.7(x)]TJ /F2 1 Tf 1.8377 0 TD [(+)-250.3(\(log)]TJ /F8 1 Tf 2.8752 0 TD (x)Tj /F2 1 Tf 0.5715 0 TD (\))Tj /F14 1 Tf 7.9701 0 0 7.9701 273.1258 571.2468 Tm (A)Tj /F2 1 Tf 10.9091 0 0 10.9091 279.9671 567.2877 Tm [(\))-375.3(con)27.8(taining)-375.2(signican)27.8(tly)-375.3(few)27.8(er)-375.3(primes)-375.3(than)]TJ -14.7737 -1.2694 TD [(usual.)-584.5(Adapting)-380(his)-380(metho)-27.8(d)-380.1(J)0(.)-380(F)83.4(riedlander)-380(and)-380(A.)-380.1(Gran)27.8(ville)-380([3])-380(sho)27.8(w)27.8(e)0(d)-380.1(that)]TJ T* [(there)-229.3(are)-229.3(arithmetic)-229.2(progressions)-229.3(con)27.8(taining)-229.2(signican)27.8(tly)-229.4(more)-229.3(\(and)-229.4(others)-229.2(with)]TJ T* [(signican)27.8(tly)-327.4(few)27.8(er\))-327.4(primes)-327.4(than)-327.4(usual.)-442.4(A)-327.4(w)27.8(eak)-327.4(form)-327.4(of)-327.4(their)-327.4(result)-327.3(is)-327.4(that,)-328.5(for)]TJ T* [(ev)27.8(ery)]TJ /F8 1 Tf 2.6387 0 TD (A)Tj /F7 1 Tf 1.0279 0 TD ()Tj /F2 1 Tf 1.0555 0 TD [(1)-330.4(there)-330.4(exist)-330.4(large)]TJ /F8 1 Tf 8.1939 0 TD (x)Tj /F2 1 Tf 0.902 0 TD [(and)-330.3(an)-330.5(arithmetic)-330.2(progression)]TJ /F8 1 Tf 13.3966 0 TD (a)Tj /F2 1 Tf 0.9731 0 TD [(\(mo)-27.8(d)]TJ /F8 1 Tf 2.6389 0 TD (q)Tj /F2 1 Tf 0.4823 0 TD [(\))-330.4(with)]TJ -31.3089 -1.2694 TD (\()Tj /F8 1 Tf 0.3889 0 TD [(a,)-166.7(q)]TJ /F2 1 Tf 1.4553 0 TD [(\))-277.8(=)-277.7(1)-333.3(and)]TJ /F8 1 Tf 4.5 0 TD (q)Tj /F7 1 Tf 0.7601 0 TD ()Tj /F8 1 Tf 1.0555 0 TD (x/)Tj /F2 1 Tf 1.0715 0 TD (\(log)Tj /F8 1 Tf 1.8472 0 TD (x)Tj /F2 1 Tf 0.5716 0 TD (\))Tj /F14 1 Tf 7.9701 0 0 7.9701 250.134 502.0065 Tm (A)Tj /F2 1 Tf 10.9091 0 0 10.9091 260.6117 498.0473 Tm [(suc)27.8(h)-333.3(that)]TJ /F9 1 Tf -2.2897 -1.0372 TD ()Tj 0 -0.6 TD ()Tj T* ()Tj /F8 1 Tf 0.3333 0.05 TD ()Tj /F2 1 Tf 0.6059 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (x)Tj /F2 1 Tf 0.5715 0 TD (;)Tj /F8 1 Tf 0.4445 0 TD 0.0359 Tc [(q,)-130.8(a)]TJ /F2 1 Tf 1.4553 0 TD 0 Tc (\))Tj /F7 1 Tf 0.6111 0 TD ()Tj /F8 1 Tf 1.1096 0.6765 TD ()Tj /F2 1 Tf 0.6059 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (x)Tj /F2 1 Tf 0.5716 0 TD (\))Tj ET 295.89 477.166 21.33 -0.436 re f BT /F8 1 Tf 10.9091 0 0 10.9091 296.3945 466.7039 Tm ()Tj /F2 1 Tf 0.5958 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (q)Tj /F2 1 Tf 0.4823 0 TD (\))Tj /F9 1 Tf 0.5481 1.836 TD ()Tj 0 -0.6 TD ()Tj T* ()Tj /F7 1 Tf 0.611 0.05 TD ()Tj /F14 1 Tf 7.9701 0 0 7.9701 335.9532 472.4981 Tm (A)Tj /F8 1 Tf 10.9091 0 0 10.9091 347.0207 481.5671 Tm ()Tj /F2 1 Tf 0.6059 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (x)Tj /F2 1 Tf 0.5715 0 TD (\))Tj ET 347.01 477.166 21.33 -0.436 re f BT /F8 1 Tf 10.9091 0 0 10.9091 347.5627 466.7039 Tm ()Tj /F2 1 Tf 0.5958 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (q)Tj /F2 1 Tf 0.4823 0 TD (\))Tj /F8 1 Tf 0.5481 0.6859 TD (.)Tj /F2 1 Tf -22.985 0 TD (\(1.1\))Tj 0 -2.1411 TD [(If)-364.1(w)27.8(e)-364.1(compare)-364.1(this)-364.1(to)-364.1(Roths)-364(b)-27.8(ound)-364.1(w)27.8(e)-364.1(note)-364.1(t)27.8(w)27.8(o)-364.1(dierences:)-506(the)-364.1(discrepancy)]TJ 0 -1.2694 TD [(exhibited)-378.8(is)-378.8(m)27.8(u)0(c)27.8(h)-378.9(larger)-378.9(in)-378.8(\(1.1\))-378.8(\(b)-27.8(eing)-378.9(within)-378.8(a)-378.9(constan)27.9(t)-378.9(factor)-378.9(of)-378.8(the)-378.8(main)]TJ T* [(term\),)-294.4(but)-284.6(the)-284.7(mo)-27.7(dulus)]TJ /F8 1 Tf 10.4624 0 TD (q)Tj /F2 1 Tf 0.7669 0 TD [(is)-284.7(m)27.8(u)0(c)27.8(h)-284.7(closer)-284.6(to)]TJ /F8 1 Tf 7.4859 0 TD (x)Tj /F2 1 Tf 0.8563 0 TD [(\(but)-284.6(not)-284.7(so)-284.7(close)-284.6(as)-284.7(to)-284.6(b)-27.8(e)-284.7(trivial\).)]TJ -17.5624 -1.2694 TD [(Recen)27.8(tly)-303.3(A.)-303.4(Balog)-303.4(and)-303.4(T.)-303.4(W)83.4(o)-27.9(oley)-303.4([1])-303.4(pro)27.8(v)27.8(e)0(d)-303.4(that)-303.4(the)-303.4(sequence)-303.3(of)-303.5(in)27.9(tegers)]TJ -2.0091 -1.2694 TD [(that)-456.2(ma)27.8(y)-456.1(b)-27.8(e)-456.2(written)-456.1(as)-456.2(the)-456.2(sum)-456.2(of)-456.3(t)27.8(w)27.8(o)-456.3(squares)-456.1(also)-456.1(exhibits)-456.1(Maier)-456.1(t)27.7(y)0(p)-27.7(e)0()]TJ T* [(irregularities)-273.8(in)-273.9(some)-273.9(in)27.9(terv)55.6(als)-273.9(\()]TJ /F8 1 Tf 13.6791 0 TD [(x,)-166.7(x)]TJ /F2 1 Tf 1.6909 0 TD [(+)-103.4(\(log)]TJ /F8 1 Tf 2.7284 0 TD (x)Tj /F2 1 Tf 0.5715 0 TD (\))Tj /F14 1 Tf 7.9701 0 0 7.9701 326.7138 385.5482 Tm (A)Tj /F2 1 Tf 10.9091 0 0 10.9091 333.5542 381.5891 Tm [(\))-273.9(for)-273.9(an)27.8(y)-274(xed,)-285.8(p)-27.7(ositiv)27.8(e)]TJ /F8 1 Tf 10.251 0 TD (A)Tj /F2 1 Tf 0.75 0 TD [(.)-424.7(While)]TJ -30.6869 -1.2694 TD [(previously)-472.9(Maiers)-473(results)-473.1(on)-473(primes)-473(had)-473.1(seemed)-473(inextricably)-473(link)27.8(ed)-473(to)-473.1(the)]TJ T* [(m)27.8(ysteries)-251.5(of)-251.6(the)-251.5(primes,)-267.8(Balog)-251.5(and)-251.5(W)83.3(o)-27.8(oleys)-251.5(example)-251.5(suggests)-251.5(that)-251.5(suc)27.8(h)-251.6(results)]TJ T* [(should)-256.1(b)-27.8(e)-256.2(part)-256.2(of)-256.2(a)-256.3(general)-256.2(phenomenon.)-418.5(Indeed,)-271.6(w)27.8(e)-256.2(will)-256.2(pro)27.8(vide)-256.1(here)-256.2(a)-256.3(general)]TJ T* [(framew)27.9(ork)-317(for)-316.9(suc)27.8(h)-316.9(results)-316.9(on)-316.9(irregularities)-316.7(of)-316.9(distribution,)-320.1(whic)27.8(h)-317(will)-316.9(include,)]TJ T* [(among)-418.5(other)-418.6(examples,)-439.8(the)-418.6(sequence)-418.6(of)-418.6(primes)-418.5(and)-418.6(the)-418.6(sequence)-418.6(of)-418.6(sums)-418.5(of)]TJ T* [(t)27.8(w)27.8(o)-375.1(squares.)-569.4(Our)-375(results)-375(ma)27.8(y)-375.1(b)-27.7(e)-375(view)27.8(ed)-375.1(as)-374.9(an)-375(uncertain)27.9(t)27.8(y)-375.1(principle)-375.1(whic)27.8(h)]TJ T* [(establishes)-453.6(that)-453.7(most)-453.7(arithmetic)-453.6(sequences)-453.7(of)-453.7(in)27.8(terest)-453.6(are)-453.7(either)-453.7(not-so-w)27.9(ell)]TJ T* [(distributed)-295(in)-295.2(longish)-295.2(arithmetic)-295.1(progressions,)-302.6(or)-295.2(are)-295.2(not-so-w)27.9(ell)-295.1(distributed)-295(in)]TJ T* [(b)-27.8(oth)-333.2(short)-333.3(in)27.8(terv)55.6(als)-333.3(and)-333.3(short)-333.2(arithmetic)-333.3(progressions.)]TJ 2.0091 -2.1826 TD (1a.)Tj /F16 1 Tf 1.7006 0 TD (Examples.)Tj /F2 1 Tf 4.8361 0 TD [(W)83.3(e)-268.5(no)27.8(w)-268.5(highligh)27.9(t)-268.6(this)-268.4(phenomenom)-268.4(with)-268.5(sev)27.8(eral)-268.4(examples:)]TJ -8.5458 -1.2694 TD [(F)83.4(o)0(r)-392.6(a)-392.6(giv)27.9(en)-392.6(set)-392.6(of)-392.6(in)27.8(tegers)]TJ /F7 1 Tf 11.9972 0 TD (A)Tj /F2 1 Tf 0.7985 0 TD [(,)-407.4(let)]TJ /F7 1 Tf 2.1889 0 TD (A)Tj /F2 1 Tf 0.7985 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (N)Tj /F2 1 Tf 0.9125 0 TD [(\))-392.6(denote)-392.5(the)-392.5(n)27.7(u)0(m)27.8(b)-27.7(er)-392.6(of)-392.6(elemen)27.8(ts)-392.5(of)]TJ /F7 1 Tf 16.0898 0 TD (A)Tj /F2 1 Tf -33.1742 -1.2694 TD [(whic)27.8(h)-333.3(are)]TJ /F7 1 Tf 4.5306 0 TD ()Tj /F8 1 Tf 1.0555 0 TD (N)Tj /F2 1 Tf 0.9125 0 TD [(,)-333.3(and)]TJ /F7 1 Tf 2.5555 0 TD (A)Tj /F2 1 Tf 0.7984 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (N)Tj /F2 1 Tf 0.9125 0 TD (;)Tj /F8 1 Tf 0.4444 0 TD 0.0359 Tc [(q,)-130.8(a)]TJ /F2 1 Tf 1.4553 0 TD 0 Tc [(\))-333.3(denote)-333.3(those)-333.2(that)-333.3(are)]TJ /F7 1 Tf 10.3973 0 TD ()Tj /F8 1 Tf 1.0555 0 TD (N)Tj /F2 1 Tf 1.2458 0 TD (and)Tj /F7 1 Tf 1.9445 0 TD ()Tj /F8 1 Tf 1.0555 0 TD (a)Tj /F2 1 Tf 0.9731 0 TD [(\(mo)-27.7(d)]TJ /F8 1 Tf 2.6388 0 TD (q)Tj /F2 1 Tf 0.4823 0 TD 0.0001 Tc (\).)Tj /F7 1 Tf -30.8374 -1.6738 TD 0 Tc ()Tj /F2 1 Tf 0.8338 0 TD [(W)83.4(e)-333.8(sa)27.8(w)-333.8(i)0(n)-333.7(Maiers)-333.8(theorem)-333.6(that)-333.7(the)-333.7(primes)-333.7(are)-333.7(not)-333.7(so)-333.8(w)27.8(ell-distributed.)]TJ -2.8429 -1.2694 TD [(W)83.3(e)-407.7(migh)27.8(t)-407.8(ask)-407.7(whether)-407.6(there)-407.7(are)-407.8(subsets)]TJ /F7 1 Tf 18.3849 0 TD (A)Tj /F2 1 Tf 1.2062 0 TD [(of)-407.8(the)-407.7(primes)-407.7(up)-407.8(to)]TJ /F8 1 Tf 9.1305 0 TD (x)Tj /F2 1 Tf 0.9793 0 TD [(whic)27.8(h)-407.8(are)]TJ -29.7009 -1.2694 TD [(w)27.8(ell-distributed.)-498.7(Fix)]TJ /F8 1 Tf 9.4003 0 TD (u)Tj /F7 1 Tf 0.8805 0 TD ()Tj /F2 1 Tf 1.0858 0 TD [(1.)-498.8(W)83.3(e)-351.4(sho)27.8(w)-351.5(that)-351.4(for)-351.5(an)27.8(y)]TJ /F8 1 Tf 11.1533 0 TD (x)Tj /F2 1 Tf 0.923 0 TD [(there)-351.4(exists)]TJ /F8 1 Tf 5.3557 0 TD (y)Tj /F7 1 Tf 0.8342 0 TD ()Tj /F2 1 Tf 0.9746 0 TD (\()Tj /F8 1 Tf 0.389 0 TD (x/)Tj /F2 1 Tf 1.0715 0 TD (4)Tj /F8 1 Tf 0.5 0 TD 0.1667 Tc (,x)Tj /F2 1 Tf 1.016 0 TD 0 Tc (\))Tj -33.5838 -1.2694 TD [(suc)27.8(h)-333.3(that)-333.3(either)]TJ 0 -1.6888 TD (\(1)Tj /F8 1 Tf 0.8889 0 TD (.)Tj /F2 1 Tf 0.2778 0 TD (2a\))Tj /F7 1 Tf 9.1218 0 TD (|A)Tj /F2 1 Tf 1.0763 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (y)Tj /F2 1 Tf 0.5262 0 TD (\))Tj /F8 1 Tf 0.3889 0 TD (/y)Tj /F7 1 Tf 1.2484 0 TD 0.2222 Tc (A)Tj /F2 1 Tf 1.7984 0 TD 0 Tc (\()Tj /F8 1 Tf 0.389 0 TD (x)Tj /F2 1 Tf 0.5715 0 TD (\))Tj /F8 1 Tf 0.3889 0 TD (/x)Tj /F7 1 Tf 1.0715 0 TD 0.2778 Tc (|)Tj /F14 1 Tf 7.9701 0 0 7.9701 333.6209 125.5861 Tm 0 Tc (u)Tj /F7 1 Tf 10.9091 0 0 10.9091 342.0523 127.2229 Tm (A)Tj /F2 1 Tf 0.7984 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (x)Tj /F2 1 Tf 0.5715 0 TD (\))Tj /F8 1 Tf 0.389 0 TD (/x)Tj ET endstream endobj 21 0 obj << /ProcSet [/PDF /Text ] /Font << /F1 4 0 R /F2 5 0 R /F5 8 0 R /F7 10 0 R /F8 11 0 R /F9 12 0 R /F14 22 0 R /F15 23 0 R /F16 24 0 R >> /ExtGState << /GS1 17 0 R >> >> endobj 26 0 obj << /Length 14111 >> stream BT /F5 1 Tf 7.9701 0 0 7.9701 165.1009 726.9739 Tm 0 g /GS1 gs 0 Tc 0 Tw [(AN)-377.7(UNCER)87.7(T)87.5(AINTY)-377.7(PRINCIPLE)-377.4(F)29.1(O)0(R)-377.6(ARITHMETIC)-377.4(SEQUENCES)]TJ /F2 1 Tf 10.9091 0 0 10.9091 483.9552 726.9741 Tm (3)Tj -33.4726 -2.3745 TD [(\(meaning)-226.9(that)-226.9(the)-226.9(subset)-227(is)-226.9(p)-27.7(o)-27.8(orly)-226.9(distributed)-226.8(in)-227(short)-226.8(in)27.8(terv)55.6(als\),)-248.2(or)-226.9(there)-226.9(exists)]TJ 0 -1.2694 TD [(some)-343.5(arithmetic)-343.4(progression)]TJ /F8 1 Tf 12.6112 0 TD (a)Tj /F2 1 Tf 0.973 0 TD [(\(mo)-27.8(d)]TJ /F8 1 Tf 2.6389 0 TD ()Tj /F2 1 Tf 0.4167 0 TD [(\))-343.5(with)-343.5(\()]TJ /F8 1 Tf 3.4093 0 TD [(a,)-166.6()]TJ /F2 1 Tf 1.3897 0 TD 0.2948 Tc [(\)=1)-48.7(a)294.8(n)294.8(d)]TJ /F8 1 Tf 4.5544 0 TD 0 Tc ()Tj /F7 1 Tf 0.7114 0 TD ()Tj /F8 1 Tf 1.0726 0 TD (x/)Tj /F2 1 Tf 1.0715 0 TD (\(log)Tj /F8 1 Tf 1.8472 0 TD (x)Tj /F2 1 Tf 0.5715 0 TD (\))Tj /F14 1 Tf 7.9701 0 0 7.9701 464.1425 691.1821 Tm (u)Tj /F2 1 Tf 10.9091 0 0 10.9091 469.5432 687.2229 Tm [(,)-346.1(for)]TJ -32.1515 -1.2694 TD [(whic)27.8(h)]TJ 0 -3.0384 TD (\(1)Tj /F8 1 Tf 0.8889 0 TD (.)Tj /F2 1 Tf 0.2778 0 TD (2b\))Tj /F9 1 Tf 9.3984 1.15 TD ()Tj 0 -0.6 TD ()Tj T* ()Tj /F7 1 Tf 0.3333 0.05 TD (A)Tj /F2 1 Tf 0.7984 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (y)Tj /F2 1 Tf 0.5262 0 TD (;)Tj /F8 1 Tf 0.4445 0 TD [(,)-166.6(a)]TJ /F2 1 Tf 1.3897 0 TD (\))Tj /F7 1 Tf 0.6111 0 TD ()Tj 1.1096 0.6765 TD (A)Tj /F2 1 Tf 0.7985 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (y)Tj /F2 1 Tf 0.5261 0 TD (\))Tj ET 295.17 643.126 22.935 -0.436 re f BT /F8 1 Tf 10.9091 0 0 10.9091 296.8668 632.7462 Tm ()Tj /F2 1 Tf 0.5959 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD ()Tj /F2 1 Tf 0.4167 0 TD (\))Tj /F9 1 Tf 0.6545 1.836 TD ()Tj 0 -0.6 TD ()Tj T* ()Tj /F7 1 Tf 0.6111 0.05 TD ()Tj /F14 1 Tf 7.9701 0 0 7.9701 336.8708 638.5933 Tm (u)Tj /F7 1 Tf 10.9091 0 0 10.9091 346.4977 647.6094 Tm (A)Tj /F2 1 Tf 0.7985 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (x)Tj /F2 1 Tf 0.5715 0 TD (\))Tj ET 346.53 643.126 23.43 -0.436 re f BT /F8 1 Tf 10.9091 0 0 10.9091 348.4483 632.7462 Tm ()Tj /F2 1 Tf 0.5958 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD ()Tj /F2 1 Tf 0.4167 0 TD (\))Tj /F8 1 Tf 0.6773 0.6859 TD (.)Tj /F2 1 Tf -23.1298 -3.2422 TD [(In)-373.2(other)-373.2(w)27.8(ords,)-383.1(w)27.7(e)-373.2(nd)-373.2(Maier)-373.1(t)27.7(y)0(p)-27.7(e)0()-373.2(irregularities)-373.1(in)-373.2(the)-373.2(distribution)-373.1(of)]TJ /F16 1 Tf 32.4137 0 TD (any)Tj -32.4137 -1.2694 TD [(subset)-360(of)-360.1(the)-360(primes)]TJ /F2 1 Tf 8.7342 0 TD [(.)-452.1(\(If)-335.8(w)27.8(e)-335.8(had)-335.9(c)27.8(hosen)]TJ /F7 1 Tf 8.7456 0 TD (A)Tj /F2 1 Tf 1.1343 0 TD [(to)-335.8(b)-27.7(e)-335.9(the)-335.8(primes)]TJ /F7 1 Tf 7.5462 0 TD ()Tj /F2 1 Tf 1.0598 0 TD [(5)-444.4(\(mo)-27.8(d)-333.3(7)0(\))-335.8(then)]TJ -27.2201 -1.2694 TD [(this)-243.2(is)-243.2(of)-243.2(no)-243.1(in)27.8(terest)-243.1(when)-243.2(w)27.8(e)-243.2(tak)27.8(e)]TJ /F8 1 Tf 14.604 0 TD (a)Tj /F2 1 Tf 0.8064 0 TD 0.2777 Tc (=1)Tj /F8 1 Tf 1.5555 0 TD 0.1667 Tc (,)Tj /F2 1 Tf 1.1389 0 TD 0 Tc [(=)-277.7(7)0(.)-414.4(T)83.3(o)-243.2(a)27.8(v)27.8(oid)-243.2(this)-243.2(minor)-243.2(tec)27.8(hnicalit)27.8(y)]TJ -18.1048 -1.2694 TD [(w)27.8(e)-316.8(can)-316.8(add)-316.8(F)83.3(or)-316.7(a)-316.9(giv)27.9(en)-316.9(nite)-316.7(set)-316.8(of)-316.8(bad)-316.8(primes)]TJ /F7 1 Tf 22.2378 0 TD (S)Tj /F2 1 Tf 0.6805 0 TD [(,)-320.1(w)27.8(e)-316.8(can)-316.8(c)27.7(h)0(o)-27.7(ose)-316.8(suc)27.8(h)-316.9(a)0(n)]TJ /F8 1 Tf 10.6375 0 TD ()Tj /F2 1 Tf -33.5559 -1.2694 TD [(for)-307.4(whic)27.8(h)-307.5(\()]TJ /F8 1 Tf 4.7288 0 TD (,)Tj /F7 1 Tf 0.8611 0 TD (S)Tj /F2 1 Tf 0.6805 0 TD [(\))-277.8(=)-277.8(1.)-435.8(Here)-307.4(and)-307.4(henceforth)-307.3(\()]TJ /F8 1 Tf 12.9749 0 TD (,)Tj /F7 1 Tf 0.8611 0 TD (S)Tj /F2 1 Tf 0.6805 0 TD [(\))-277.8(=)-277.8(1)-307.5(means)-307.3(that)-307.4(\()]TJ /F8 1 Tf 8.0945 0 TD [(,)-166.6(p)]TJ /F2 1 Tf 1.3643 0 TD [(\))-277.8(=)-277.7(1)-307.4(for)]TJ -30.2458 -1.2694 TD (all)Tj /F8 1 Tf 1.3889 0 TD (p)Tj /F7 1 Tf 0.7809 0 TD 0.2778 Tc (S)Tj /F2 1 Tf 1.625 0 TD 0 Tc (.\))Tj /F7 1 Tf -1.7857 -2.3236 TD ()Tj /F2 1 Tf 0.8795 0 TD [(With)-379.4(probabilit)27.9(y)-379.4(1)-379.5(there)-379.4(are)]TJ /F16 1 Tf 12.9891 0 TD (no)Tj /F2 1 Tf 1.4529 0 TD [(Maier)-379.4(t)27.8(y)0(p)-27.8(e)0.1()-379.5(irregularities)-379.3(in)-379.5(the)-379.5(dis-)]TJ -17.3307 -1.2694 TD [(tribution)-389.3(of)-389.3(randomly)-389.3(c)27.7(hosen)-389.3(subsets)-389.3(of)-389.3(the)-389.4(in)27.9(tegers.)-612.5(Indeed)-389.4(suc)27.8(h)-389.4(irregulari-)]TJ T* [(ties)-405.7(seem)-405.8(to)-405.8(dep)-27.7(end)-405.8(on)-405.7(the)-405.8(subset)-405.7(ha)27.8(ving)-405.8(some)-405.7(arithmetic)-405.7(structure.)-661.6(So)-405.8(in-)]TJ T* [(stead)-437.3(of)-437.4(taking)-437.2(subsets)-437.3(of)-437.4(all)-437.3(the)-437.4(in)27.9(tegers,)-463.4(w)27.8(e)-437.4(need)-437.3(to)-437.3(tak)27.7(e)-437.3(subsets)-437.3(of)-437.3(a)-437.4(set)]TJ T* [(whic)27.8(h)-432(already)-431.9(has)-432(some)-431.9(arithmetic)-431.9(structure.)-740.3(F)83.4(o)0(r)-432(example,)-456.6(dene)]TJ /F7 1 Tf 30.1823 0 TD (S)Tj /F14 1 Tf 7.9701 0 0 7.9701 454.667 453.2433 Tm ()Tj /F2 1 Tf 10.9091 0 0 10.9091 463.7878 454.8792 Tm [(to)-432(b)-27.7(e)]TJ -31.6239 -1.2694 TD [(the)-344.7(set)-344.8(of)-344.7(in)27.8(tegers)]TJ /F8 1 Tf 8.1707 0 TD (n)Tj /F2 1 Tf 0.9449 0 TD [(ha)27.7(ving)-344.7(no)-344.7(prime)-344.7(factors)-344.6(in)-344.7(the)-344.8(in)27.8(terv)55.6(al)-344.8([\(log)]TJ /F8 1 Tf 19.4133 0 TD (n)Tj /F2 1 Tf 0.6002 0 TD (\))Tj /F1 1 Tf 7.9701 0 0 7.9701 440.815 444.9903 Tm (1)Tj /F15 1 Tf 0.5312 0 TD ()Tj /F14 1 Tf 0.8264 0 TD ()Tj /F8 1 Tf 10.9091 0 0 10.9091 456.0439 441.0311 Tm (,)Tj /F2 1 Tf 0.4444 0 TD (log)Tj /F8 1 Tf 1.4584 0 TD (n)Tj /F2 1 Tf 0.6003 0 TD -0.0001 Tc (],)Tj -33.4171 -1.2694 TD 0 Tc [(so)-406.9(that)]TJ /F7 1 Tf 3.5416 0 TD (S)Tj /F14 1 Tf 7.9701 0 0 7.9701 164.0408 425.5472 Tm ()Tj /F2 1 Tf 10.9091 0 0 10.9091 168.4493 427.183 Tm (\()Tj /F8 1 Tf 0.389 0 TD (N)Tj /F2 1 Tf 0.9124 0 TD (\))Tj /F7 1 Tf 0.7893 0 TD ()Tj /F2 1 Tf 1.1782 0 TD (\(1)Tj /F7 1 Tf 1.1601 0 TD ()Tj /F8 1 Tf 1.049 0 TD ()Tj /F2 1 Tf 0.4663 0 TD (\))Tj /F8 1 Tf 0.389 0 TD (N)Tj /F2 1 Tf 0.9125 0 TD [(.)-665.1(Notice)-406.9(that)-406.8(the)-406.9(primes)-406.8(are)-406.9(a)-406.9(subset)-406.8(of)]TJ /F7 1 Tf 18.498 0 TD (S)Tj /F14 1 Tf 7.9701 0 0 7.9701 455.8964 425.5472 Tm ()Tj /F2 1 Tf 10.9091 0 0 10.9091 460.3049 427.183 Tm [(.)-665.1(Our)]TJ -31.3047 -1.2694 TD [(results)-445(imply)-445(that)-445.1(an)27.8(y)-445.1(subset)]TJ /F7 1 Tf 13.667 0 TD (A)Tj /F2 1 Tf 1.2436 0 TD (of)Tj /F7 1 Tf 1.2507 0 TD (S)Tj /F14 1 Tf 7.9701 0 0 7.9701 301.7106 411.6991 Tm ()Tj /F2 1 Tf 10.9091 0 0 10.9091 310.9738 413.335 Tm [(is)-445.1(p)-27.7(o)-27.8(orly)-445.1(distributed)-444.9(in)-445.1(that)-445(for)-445.1(an)27.8(y)]TJ /F8 1 Tf -17.616 -1.2694 TD (x)Tj /F2 1 Tf 1.1139 0 TD [(there)-542.3(exists)]TJ /F8 1 Tf 5.7375 0 TD (y)Tj /F7 1 Tf 1.1523 0 TD ()Tj /F2 1 Tf 1.2929 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (x/)Tj /F2 1 Tf 1.0715 0 TD (4)Tj /F8 1 Tf 0.5 0 TD 0.1667 Tc (,x)Tj /F2 1 Tf 1.016 0 TD 0 Tc [(\))-542.4(suc)27.8(h)-542.3(that)-542.3(either)-542.3(\(1.2a\))-542.4(holds,)-594.5(or)-542.4(there)-542.3(exists)]TJ -12.273 -1.2694 TD [(some)-456.4(arithmetic)-456.4(progression)]TJ /F8 1 Tf 12.95 0 TD (a)Tj /F2 1 Tf 0.9731 0 TD [(\(mo)-27.7(d)]TJ /F8 1 Tf 2.6389 0 TD ()Tj /F2 1 Tf 0.4166 0 TD [(\))-456.5(and)]TJ /F8 1 Tf 2.913 0 TD ()Tj /F7 1 Tf 0.8996 0 TD ()Tj /F8 1 Tf 1.2608 0 TD (x/)Tj /F2 1 Tf 1.0715 0 TD (\(log)Tj /F8 1 Tf 1.8472 0 TD (x)Tj /F2 1 Tf 0.5715 0 TD (\))Tj /F14 1 Tf 7.9701 0 0 7.9701 401.6847 389.5981 Tm (u)Tj /F2 1 Tf 10.9091 0 0 10.9091 412.0648 385.6389 Tm [(with)-456.4(\()]TJ /F8 1 Tf 2.7899 0 TD [(a,)-166.6()]TJ /F2 1 Tf 1.3897 0 TD 0.483 Tc [(\)=1)483(,)]TJ -31.0622 -1.2694 TD 0 Tc [(for)-527.2(whic)27.8(h)-527.2(a)-527.2(suitably)-527.2(mo)-27.7(died)-527.2(\(1.2b\))-527.2(holds)-527.1(\(that)-527.2(is)-527.2(with)]TJ /F8 1 Tf 26.4585 0 TD ()Tj /F2 1 Tf 0.5958 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD ()Tj /F2 1 Tf 0.4167 0 TD [(\))-527.2(replaced)-527.2(b)27.8(y)]TJ /F8 1 Tf -27.8598 -1.2694 TD ()Tj /F9 1 Tf 0.5834 0.7501 TD ()Tj /F14 1 Tf 7.9701 0 0 7.9701 135.466 354.7178 Tm (p)Tj /F15 1 Tf 0.5349 0 TD (|)Tj /F14 1 Tf 0.2951 0 TD (,)Tj /F1 1 Tf 1.1855 0 TD (\(log)Tj /F14 1 Tf 1.9626 0 TD (x)Tj /F1 1 Tf 0.5981 0 TD (\))Tj /F12 1 Tf 4.9813 0 0 4.9813 175.2329 356.9863 Tm (1)Tj /F11 1 Tf 0.6804 0 TD ()Tj /F10 1 Tf 1.0834 0 TD ()Tj /F14 1 Tf 7.9701 0 0 7.9701 187.8775 354.7178 Tm (
)Tj
/F2 1 Tf
1.0555 0 TD
[(1.)-441.1(Let)]TJ
/F8 1 Tf
3.0006 0 TD
(R)Tj
/F2 1 Tf
1.0904 0 TD
[(denote)-323.3(the)]TJ
-29.3715 -1.2694 TD
[(ring)-406.1(of)-406.1(in)27.8(tegers)-406(of)]TJ
/F8 1 Tf
8.33 0 TD
(K)Tj
/F2 1 Tf
1.3269 0 TD
[(and)-406.1(let)]TJ
/F8 1 Tf
3.5345 0 TD
(C)Tj
/F2 1 Tf
1.1923 0 TD
[(b)-27.7(e)-406.1(an)-406.1(ideal)-406.1(class)-406.1(from)-406(the)-406.2(class)-406(group)-406.1(of)]TJ
/F8 1 Tf
18.544 0 TD
(R)Tj
/F2 1 Tf
0.7669 0 TD
(.)Tj
-33.6948 -1.2694 TD
-0.0833 Tc
[(Ta)-83.3(k)-55.5(e)]TJ
/F7 1 Tf
2.3779 0 TD
0 Tc
(A)Tj
/F2 1 Tf
1.0931 0 TD
[(to)-294.6(b)-27.7(e)-294.6(the)-294.5(set)-294.6(of)-294.6(p)-27.7(ositiv)27.8(e)-294.5(i)0(n)27.8(tegers)-294.5(whic)27.8(h)-294.6(are)-294.5(the)-294.6(norm)-294.5(of)-294.6(some)-294.5(\(in)27.8(tegral\))]TJ
-3.471 -1.2694 TD
[(ideal)-275.3(b)-27.7(elonging)-275.3(to)]TJ
/F8 1 Tf
7.9648 0 TD
(C)Tj
/F2 1 Tf
0.7862 0 TD
[(.)-425.1(\(In)-275.2(Balog)-275.3(and)-275.2(W)83.4(o)-27.9(oleys)-275.3(example,)]TJ
/F7 1 Tf
15.2493 0 TD
(A)Tj
/F2 1 Tf
1.0738 0 TD
[(is)-275.3(the)-275.3(set)-275.2(of)-275.3(n)27.8(u)0(m)27.8(b)-27.8(ers)]TJ
-25.0742 -1.2694 TD
[(of)-296.7(the)-296.7(form)]TJ
/F8 1 Tf
5.1153 0 TD
(x)Tj
/F1 1 Tf
7.9701 0 0 7.9701 180.8379 281.1616 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 187.196 277.2035 Tm
(+)Tj
/F8 1 Tf
0.9268 0 TD
(y)Tj
/F1 1 Tf
7.9701 0 0 7.9701 203.0476 281.1616 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 207.7798 277.2035 Tm
[(,)-304(with)]TJ
/F8 1 Tf
2.823 0 TD
(C)Tj
/F2 1 Tf
1.083 0 TD
[(the)-296.7(class)-296.7(of)-296.8(principal)-296.7(ideals)-296.7(in)]TJ
/F8 1 Tf
13.1056 0 TD
(R)Tj
/F2 1 Tf
1.0448 0 TD
(=)Tj
/F17 1 Tf
1.0555 0 TD
(Z)Tj
/F2 1 Tf
0.6667 0 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
(i)Tj
/F2 1 Tf
0.3445 0 TD
[(].\))-432.2(F)83.3(rom)-296.7(our)]TJ
-28.5574 -1.2694 TD
[(w)27.8(ork)-317.8(it)-317.7(follo)27.8(ws)-317.7(that)-317.7(the)-317.7(set)]TJ
/F7 1 Tf
12.087 0 TD
(A)Tj
/F2 1 Tf
1.1163 0 TD
[(is)-317.7(p)-27.8(o)-27.8(orly)-317.7(distributed)-317.6(in)-317.7(arithmetic)-317.6(progressions;)]TJ
-13.2033 -1.2694 TD
[(that)-415.3(is,)-435.8(a)-415.3(suitably)-415.3(mo)-27.7(died)-415.4(v)27.8(ersion)-415.3(of)-415.3(\(1.2b\))-415.3(holds.)-690.4(Moreo)27.8(v)27.8(er,)-435.9(if)-415.3(w)27.7(e)-415.3(replace)]TJ
/F8 1 Tf
T*
(R)Tj
/F2 1 Tf
1.1661 0 TD
[(b)27.8(y)-399.2(an)27.8(y)-399.1(order)-399.1(in)]TJ
/F8 1 Tf
7.3242 0 TD
(K)Tj
/F2 1 Tf
1.32 0 TD
[(then)-399(either)-399.1(\(1.2a\))-399.1(holds)-399.1(or)-399.1(a)-399.1(suitably)-399(mo)-27.7(died)-399.1(v)27.8(ersion)]TJ
-9.8103 -1.2694 TD
[(of)-349.4(\(1.2b\))-349.5(holds)-349.4(\(and)-349.4(w)27.8(e)-349.4(exp)-27.8(ect)-349.4(that,)-353.4(with)-349.4(some)-349.5(eort,)-353.4(one)-349.5(can)-349.4(pro)27.8(v)27.8(e)-349.4(that)-349.5(the)]TJ
T*
[(suitably)-333.3(mo)-27.7(died)-333.3(\(1.2b\))-333.3(holds\).)]TJ
/F7 1 Tf
2.0091 -2.3236 TD
()Tj
/F2 1 Tf
0.7907 0 TD
(Let)Tj
/F7 1 Tf
1.7489 0 TD
(B)Tj
/F2 1 Tf
0.9779 0 TD
[(b)-27.7(e)-290.7(a)-290.6(giv)27.8(en)-290.7(set)-290.6(of)]TJ
/F8 1 Tf
7.2921 0 TD
(x)Tj
/F2 1 Tf
0.8622 0 TD
[(in)27.8(tegers)-290.6(and)]TJ
/F7 1 Tf
5.5619 0 TD
(P)Tj
/F2 1 Tf
1.0684 0 TD
[(b)-27.8(e)-290.6(a)-290.7(giv)27.9(en)-290.7(set)-290.6(of)-290.6(primes.)-430.1(Dene)]TJ
/F7 1 Tf
-20.3113 -1.2694 TD
(S)Tj
/F2 1 Tf
0.6805 0 TD
(\()Tj
/F7 1 Tf
0.3889 0 TD
(B)Tj
/F8 1 Tf
0.6873 0 TD
(,)Tj
/F7 1 Tf
0.4444 0 TD
(P)Tj
/F8 1 Tf
0.7778 0 TD
0.1666 Tc
(,z)Tj
/F2 1 Tf
0.9534 0 TD
0 Tc
[(\))-320.6(t)0(o)-320.6(b)-27.7(e)-320.6(the)-320.6(n)27.8(u)0(m)27.8(b)-27.7(er)-320.6(of)-320.5(in)27.8(tegers)-320.5(in)]TJ
/F7 1 Tf
14.5757 0 TD
(B)Tj
/F2 1 Tf
1.0078 0 TD
[(whic)27.8(h)-320.6(d)0(o)-320.5(not)-320.5(ha)27.7(v)27.8(e)-320.5(a)-320.6(prime)-320.5(factor)]TJ
/F8 1 Tf
-19.5158 -1.2694 TD
(p)Tj
/F7 1 Tf
0.7889 0 TD
0.2857 Tc
(P)Tj
/F2 1 Tf
2.0682 0 TD
0 Tc
(with)Tj
/F8 1 Tf
2.2826 0 TD
(p)Tj
/F7 1 Tf
0.7889 0 TD
()Tj
/F8 1 Tf
1.0635 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
[(.)-458.7(Siev)27.8(e)-338.1(theory)-338(is)-338.1(concerned)-338(with)-338.1(estimating)]TJ
/F7 1 Tf
19.3095 0 TD
(S)Tj
/F2 1 Tf
0.6805 0 TD
(\()Tj
/F7 1 Tf
0.3889 0 TD
(B)Tj
/F8 1 Tf
0.6872 0 TD
(,)Tj
/F7 1 Tf
0.4445 0 TD
(P)Tj
/F8 1 Tf
0.7778 0 TD
0.1666 Tc
(,z)Tj
/F2 1 Tf
0.9534 0 TD
0 Tc
[(\))-338.1(under)]TJ
-30.7428 -1.2694 TD
[(certain)-427.2(natural)-427.1(h)27.8(y)0(p)-27.8(otheses)-427.1(for)]TJ
/F7 1 Tf
13.8395 0 TD
(B)Tj
/F8 1 Tf
0.6872 0 TD
(,)Tj
/F7 1 Tf
0.4445 0 TD
(P)Tj
/F2 1 Tf
1.2049 0 TD
(and)Tj
/F8 1 Tf
2.0384 0 TD
(u)Tj
/F2 1 Tf
1.0068 0 TD
[(:=)-434.2(log)]TJ
/F8 1 Tf
2.9481 0 TD
(x/)Tj
/F2 1 Tf
1.2382 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.5089 0 TD
[(.)-726.1(The)-427.2(fundamen)27.9(tal)]TJ
-25.3748 -1.2694 TD
[(lemma)-242.1(of)-242.2(siev)27.9(e)-242.2(theory)-242.1(\(see)-242.1([7]\))-242.2(implies)-242.1(\(for)-242.2(example)-242.1(when)]TJ
/F7 1 Tf
24.6106 0 TD
(B)Tj
/F2 1 Tf
0.9294 0 TD
[(is)-242.2(the)-242.1(set)-242.2(of)-242.2(in)27.9(tegers)]TJ
ET
endstream
endobj
27 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F10 13 0 R
/F11 14 0 R
/F12 15 0 R
/F14 22 0 R
/F15 23 0 R
/F16 24 0 R
/F17 28 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
30 0 obj
<<
/Length 17590
>>
stream
BT
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
(4)Tj
/F5 1 Tf
7.9701 0 0 7.9701 199.3913 726.9739 Tm
[(ANDREW)-377.5(GRANVILLE)-377.5(AND)-377.6(K)-0.1(.)-377.7(SOUND)29.3(ARARAJAN)]TJ
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 701.071 Tm
[(in)-333.3(an)-333.3(in)27.8(terv)55.6(al\))-333.3(that)]TJ
/F9 1 Tf
2.2488 -0.8067 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
T*
()Tj
T*
()Tj
T*
()Tj
/F7 1 Tf
0.3333 0.9501 TD
(S)Tj
/F2 1 Tf
0.6806 0 TD
(\()Tj
/F7 1 Tf
0.3889 0 TD
(B)Tj
/F8 1 Tf
0.6872 0 TD
(,)Tj
/F7 1 Tf
0.4445 0 TD
(P)Tj
/F8 1 Tf
0.7777 0 TD
0.1667 Tc
(,z)Tj
/F2 1 Tf
0.9535 0 TD
0 Tc
(\))Tj
/F7 1 Tf
0.611 0 TD
()Tj
/F8 1 Tf
1 0 TD
(x)Tj
/F9 1 Tf
1.6564 0.9499 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 215.4959 656.6436 Tm
(p)Tj
/F15 1 Tf
0.5347 0 TD
(P)Tj
/F14 1 Tf
1.5348 0 TD
(,p)Tj
/F15 1 Tf
0.83 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F9 1 Tf
10.9091 0 0 10.9091 251.2857 685.2892 Tm
()Tj
/F2 1 Tf
0.7362 -1.41 TD
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F2 1 Tf
1.1111 0.6765 TD
(1)Tj
ET
279.33 672.886 5.488 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 279.2996 662.4249 Tm
(p)Tj
/F9 1 Tf
0.6127 2.0959 TD
()Tj
0.7361 0.64 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
T*
()Tj
T*
()Tj
T*
()Tj
/F7 1 Tf
0.6111 0.9501 TD
()Tj
/F9 1 Tf
1.2778 1.41 TD
()Tj
/F2 1 Tf
0.8457 -0.7334 TD
0.2222 Tc
(1+)Tj
/F8 1 Tf
1.7222 0 TD
0 Tc
(o)Tj
/F2 1 Tf
0.4847 0 TD
(\(1\))Tj
ET
323.85 672.886 38.015 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 327.745 662.4249 Tm
(u)Tj
/F2 1 Tf
0.7391 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(u)Tj
/F9 1 Tf
1.0395 2.0959 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 371.0864 682.5714 Tm
(u)Tj
/F8 1 Tf
10.9091 0 0 10.9091 378.3064 669.9079 Tm
(x)Tj
/F9 1 Tf
1.6563 0.95 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 386.3592 656.6436 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
-0.0001 Tc
(P)Tj
/F14 1 Tf
1.5346 0 TD
0.0001 Tc
(,p)Tj
/F15 1 Tf
0.83 0 TD
0 Tc
()Tj
/F14 1 Tf
0.8265 0 TD
(z)Tj
/F9 1 Tf
10.9091 0 0 10.9091 422.149 685.2892 Tm
()Tj
/F2 1 Tf
0.7362 -1.41 TD
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F2 1 Tf
1.1111 0.6765 TD
(1)Tj
ET
450.21 672.886 5.488 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 450.1629 662.4249 Tm
(p)Tj
/F9 1 Tf
0.6127 2.0959 TD
()Tj
/F2 1 Tf
-30.9877 -4.4275 TD
(for)Tj
/F8 1 Tf
1.6349 0 TD
0.4516 Tc
(u >
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
51 0 obj
<<
/Length 22271
>>
stream
BT
/F5 1 Tf
7.9701 0 0 7.9701 162.3741 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
[(AN)-377.7(UNCER)87.7(T)87.5(AINTY)-377.5(PRINCIPLE)-377.6(F)29.2(O)0(R)-377.8(ARITHMETIC)-377.4(SEQUENCES)]TJ
/F2 1 Tf
10.9091 0 0 10.9091 478.5006 726.9741 Tm
(11)Tj
/F5 1 Tf
-30.9635 -2.3745 TD
[(Pr)22.3(oposition)-385(2.1)]TJ
/F2 1 Tf
7.8627 0 TD
(.)Tj
/F16 1 Tf
1.2082 0 TD
[(Supp)51.2(ose)-364.2(that)]TJ
/F8 1 Tf
5.8649 0 TD
(q)Tj
/F7 1 Tf
0.7716 0 TD
()Tj
/F8 1 Tf
1.067 0 TD
(x)Tj
/F16 1 Tf
0.9357 0 TD
[(is)-364.1(an)-364.2(inte)51.2(ger)-364.2(for)-364.1(which)]TJ
/F8 1 Tf
10.254 0 TD
(h)Tj
/F2 1 Tf
0.5762 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
/F8 1 Tf
0.6782 0 TD
(>)Tj
/F2 1 Tf
1.0671 0 TD
(9)Tj
/F16 1 Tf
0.5 0 TD
(.)Tj
-33.6659 -1.2694 TD
[(Then)-357.7(either)]TJ
/F9 1 Tf
8.4533 -0.3733 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F7 1 Tf
0.3333 0.05 TD
(A)Tj
/F2 1 Tf
0.7985 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(;)Tj
/F8 1 Tf
0.4445 0 TD
0.0359 Tc
(q,)Tj
/F2 1 Tf
0.9268 0 TD
0 Tc
(0\))Tj
/F7 1 Tf
1.1111 0 TD
()Tj
/F8 1 Tf
1.1096 0.6821 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 278.3671 676.4085 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 282.932 678.0453 Tm
(\(0\))Tj
ET
272.97 673.606 23.846 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 277.2115 663.1213 Tm
0.036 Tc
(q)Tj
/F14 1 Tf
7.9701 0 0 7.9701 288.1216 661.4845 Tm
0 Tc
(q)Tj
/F7 1 Tf
10.9091 0 0 10.9091 298.0672 670.6042 Tm
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5716 0 TD
(\))Tj
/F9 1 Tf
0.3889 1.1501 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F7 1 Tf
0.6111 0.05 TD
()Tj
/F2 1 Tf
1.1651 0.6765 TD
(1)Tj
ET
340.89 673.606 5.455 -0.436 re
f
BT
10.9091 0 0 10.9091 340.8747 663.1213 Tm
(2)Tj
/F8 1 Tf
0.7192 1.368 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 354.0612 676.4085 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 358.6262 678.0453 Tm
(\(0\))Tj
ET
1 i
348.69 673.17 23.846 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 352.9056 663.1213 Tm
0.0359 Tc
(q)Tj
/F14 1 Tf
7.9701 0 0 7.9701 363.8147 661.4845 Tm
0 Tc
(q)Tj
/F7 1 Tf
10.9091 0 0 10.9091 373.7614 670.6042 Tm
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F16 1 Tf
-25.1303 -2.3318 TD
(or)Tj
/F2 1 Tf
0.9328 0 TD
(,)Tj
/F16 1 Tf
0.7663 0 TD
[(for)-462.4(every)-462.2(prime)]TJ
/F8 1 Tf
7.431 0 TD
()Tj
/F16 1 Tf
0.879 0 TD
[(in)-462.3(the)-462.4(r)51.2(ange)]TJ
/F8 1 Tf
5.9232 0 TD
(x)Tj
/F7 1 Tf
1.0388 0 TD
()Tj
/F8 1 Tf
1.245 0 TD
()Tj
/F7 1 Tf
0.8839 0 TD
()Tj
/F2 1 Tf
1.245 0 TD
(3\()Tj
/F8 1 Tf
0.8889 0 TD
(x)Tj
/F2 1 Tf
0.8696 0 TD
0.298 Tc
(+2)Tj
/F8 1 Tf
1.5758 0 TD
0 Tc
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/h)Tj
/F2 1 Tf
1.0762 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
/F16 1 Tf
0.8513 0 TD
[(which)-462.3(do)51.2(es)-462.3(not)]TJ
-27.349 -1.2694 TD
(divide)Tj
/F8 1 Tf
2.9133 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(,)Tj
/F16 1 Tf
0.6355 0 TD
[(ther)51.1(e)-357.8(i)0(s)-357.8(a)0(n)-357.7(arithmetic)-357.6(pr)51.1(o)51.1(g)0(r)51.2(ession)]TJ
/F8 1 Tf
15.001 0 TD
(b)Tj
/F2 1 Tf
0.8736 0 TD
[(\(mo)-27.8(d)]TJ
/F8 1 Tf
2.6389 0 TD
()Tj
/F2 1 Tf
0.4166 0 TD
(\))Tj
/F16 1 Tf
0.7467 0 TD
[(such)-357.7(that)]TJ
/F9 1 Tf
-15.1987 -1.1714 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F7 1 Tf
0.3333 0.05 TD
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(;)Tj
/F8 1 Tf
0.4444 0 TD
[(,)-166.7(b)]TJ
/F2 1 Tf
1.2903 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1.1096 0.6765 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 277.4884 611.5966 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 281.4456 613.3739 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(b)Tj
/F2 1 Tf
0.4292 0 TD
(\))Tj
ET
272.13 608.926 22.465 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 276.3039 598.5106 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 286.4977 596.7333 Tm
()Tj
/F7 1 Tf
10.9091 0 0 10.9091 295.8077 605.9935 Tm
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F9 1 Tf
0.3889 1.15 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F7 1 Tf
0.6111 0.05 TD
()Tj
/F2 1 Tf
1.1651 0.6765 TD
(1)Tj
ET
338.61 608.926 5.455 -0.436 re
f
BT
10.9091 0 0 10.9091 338.6152 598.5106 Tm
(2)Tj
/F8 1 Tf
0.7192 1.3625 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 351.8018 611.5966 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 355.7589 613.3739 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(b)Tj
/F2 1 Tf
0.4291 0 TD
(\))Tj
ET
346.41 608.926 22.465 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 350.6182 598.5106 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 360.811 596.7333 Tm
()Tj
/F7 1 Tf
10.9091 0 0 10.9091 370.1211 605.9935 Tm
(A)Tj
/F2 1 Tf
0.7985 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(.)Tj
/F2 1 Tf
-23.1764 -3.3031 TD
[(The)-222.2(rst)-222.2(criterion)-222.1(is)-222.2(equiv)55.6(alen)27.9(t)-222.3(t)0(o)]TJ
/F7 1 Tf
14.4362 0 TD
-0.0001 Tc
(|A)Tj
/F14 1 Tf
7.9701 0 0 7.9701 309.9437 568.3238 Tm
0 Tc
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 314.5086 569.9597 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.7777 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(h)Tj
/F2 1 Tf
0.5761 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/q)Tj
/F2 1 Tf
0.9823 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
(A)Tj
/F2 1 Tf
0.7985 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5716 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
0.2777 Tc
(|)Tj
/F1 1 Tf
7.9701 0 0 7.9701 419.1462 574.2546 Tm
0 Tc
(1)Tj
ET
419.13 572.926 4.234 -0.436 re
f
BT
7.9701 0 0 7.9701 419.1462 566.1978 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 424.5768 569.9597 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(h)Tj
/F2 1 Tf
0.5762 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
/F8 1 Tf
0.389 0 TD
(/q)Tj
/F2 1 Tf
0.9823 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(A)Tj
/F2 1 Tf
0.7985 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\),)Tj
-33.3848 -1.2694 TD
(since)Tj
/F8 1 Tf
2.45 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 150.8682 554.4757 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 155.4331 556.1116 Tm
(\(0\))Tj
/F8 1 Tf
1.2777 0 TD
[(/q)-35.9()]TJ
/F14 1 Tf
7.9701 0 0 7.9701 185.7355 554.4757 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 193.331 556.1116 Tm
(=)Tj
/F8 1 Tf
1.0555 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 210.1868 554.4757 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 214.7517 556.1116 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
[(/q)-35.9()]TJ
/F14 1 Tf
7.9701 0 0 7.9701 244.8618 554.4757 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 252.4573 556.1116 Tm
(=)Tj
/F8 1 Tf
1.0555 0 TD
(h)Tj
/F2 1 Tf
0.5762 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/q)Tj
/F2 1 Tf
0.9823 0 TD
(.)Tj
/F16 1 Tf
-14.1169 -2.1827 TD
[(Pr)51.2(o)51.1(o)0(f)]TJ
/F2 1 Tf
2.3267 0 TD
[(.)-888.9(I)0(f)-333.3(the)-333.3(rst)-333.3(option)-333.3(fails)-333.3(then)]TJ
/F9 1 Tf
-2.3897 -1.3214 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 135.5128 493.7265 Tm
(n)Tj
/F15 1 Tf
0.6446 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(x/q)Tj
/F7 1 Tf
10.9091 0 0 10.9091 162.123 507.5218 Tm
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(;)Tj
/F8 1 Tf
0.4445 0 TD
[(,)-166.6(nq)]TJ
/F2 1 Tf
1.9437 0 TD
(\))Tj
/F7 1 Tf
0.6666 0 TD
()Tj
/F9 1 Tf
1.4697 0.95 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 226.1509 493.7265 Tm
(n)Tj
/F15 1 Tf
0.6446 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(x/q)Tj
/F8 1 Tf
10.9091 0 0 10.9091 252.7611 507.5218 Tm
(a)Tj
/F2 1 Tf
0.5286 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(nq)Tj
/F2 1 Tf
1.0825 0 TD
0.2778 Tc
(\)=)Tj
/F7 1 Tf
1.7223 0 TD
0 Tc
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(;)Tj
/F8 1 Tf
0.4445 0 TD
0.0359 Tc
(q,)Tj
/F2 1 Tf
0.9268 0 TD
0 Tc
(0\))Tj
/F7 1 Tf
1.1667 0 TD
()Tj
/F2 1 Tf
1.1651 0.6765 TD
(1)Tj
ET
353.01 510.526 5.455 -0.436 re
f
BT
10.9091 0 0 10.9091 352.9514 500.0389 Tm
(2)Tj
/F8 1 Tf
0.7192 1.368 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 366.138 513.326 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 370.7029 514.9629 Tm
(\(0\))Tj
ET
360.81 510.09 23.846 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 364.9823 500.0389 Tm
0.0359 Tc
(q)Tj
/F14 1 Tf
7.9701 0 0 7.9701 375.8924 498.403 Tm
0 Tc
(q)Tj
/F7 1 Tf
10.9091 0 0 10.9091 385.8381 507.5218 Tm
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5716 0 TD
0.2778 Tc
(\)=)Tj
/F8 1 Tf
1.8318 0.6765 TD
0 Tc
(h)Tj
/F2 1 Tf
0.5762 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
ET
425.01 510.526 20.032 -0.436 re
f
BT
10.9091 0 0 10.9091 429.6668 500.0389 Tm
(2)Tj
/F8 1 Tf
0.5 0 TD
(q)Tj
/F7 1 Tf
1.0189 0.6859 TD
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F8 1 Tf
0.389 0 TD
(.)Tj
/F2 1 Tf
-32.1628 -2.9422 TD
[(On)-341(the)-340.9(other)-340.9(hand,)-342.8(if)-341(prime)]TJ
/F8 1 Tf
12.581 0 TD
()Tj
/F17 1 Tf
0.7071 0 TD
()Tj
/F8 1 Tf
0.5683 0 TD
(q)Tj
/F2 1 Tf
0.8233 0 TD
(then)Tj
/F8 1 Tf
2.2854 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 309.2144 473.6479 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 313.1716 475.4252 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(nq)Tj
/F2 1 Tf
1.0826 0 TD
0.2905 Tc
[(\)=1)-50.4(i)290.5(f)]TJ
/F8 1 Tf
3.5129 0 TD
0 Tc
()Tj
/F17 1 Tf
0.7071 0 TD
()Tj
/F8 1 Tf
0.5683 0 TD
(n)Tj
/F2 1 Tf
0.6002 0 TD
[(,)-343(and)]TJ
/F8 1 Tf
2.5728 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 421.4147 473.6479 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 425.3719 475.4252 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(nq)Tj
/F2 1 Tf
1.0826 0 TD
0.2905 Tc
(\)=)Tj
/F8 1 Tf
1.7477 0 TD
0 Tc
(h)Tj
/F2 1 Tf
0.5761 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4166 0 TD
(\))Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 485.4526 473.6479 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 461.5771 Tm
(if)Tj
/F8 1 Tf
0.9167 0 TD
()Tj
/F7 1 Tf
0.4166 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(n)Tj
/F2 1 Tf
0.6002 0 TD
[(.)-444.5(Therefore)-333.2(for)-333.3(an)27.8(y)]TJ
/F8 1 Tf
8.6751 0 TD
(N)Tj
/F2 1 Tf
0.9125 0 TD
(,)Tj
/F9 1 Tf
-5.8937 -1.3714 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 181.4516 422.9888 Tm
(n)Tj
/F15 1 Tf
0.6446 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(N)Tj
/F8 1 Tf
10.9091 0 0 10.9091 203.7599 443.6334 Tm
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 209.1009 441.856 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 213.058 443.6334 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(nq)Tj
/F2 1 Tf
1.0825 0 TD
(\))Tj
ET
203.73 439.246 29.593 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 211.4809 428.7701 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 221.6747 426.9928 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 236.3666 436.253 Tm
(=)Tj
/F9 1 Tf
1.0508 0.95 TD
()Tj
/F10 1 Tf
4.9813 0 0 4.9813 247.8655 424.8518 Tm
(n)Tj
/F11 1 Tf
0.8808 0 TD
()Tj
/F10 1 Tf
1.0834 0 TD
(N)Tj
-1.2994 -1.3594 TD
(l)Tj
/F18 1 Tf
0.4802 0 TD
()Tj
/F10 1 Tf
0.4584 0 TD
(n)Tj
/F2 1 Tf
10.9091 0 0 10.9091 272.1086 443.6334 Tm
(1)Tj
ET
267.81 439.246 14.151 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 267.7599 428.7701 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 277.9537 426.9928 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 285.5313 436.253 Tm
(+)Tj
/F9 1 Tf
1.1062 0.95 TD
()Tj
/F10 1 Tf
4.9813 0 0 4.9813 297.6359 424.8518 Tm
(n)Tj
/F11 1 Tf
0.8808 0 TD
()Tj
/F10 1 Tf
1.0832 0 TD
(N)Tj
-1.2994 -1.3594 TD
(l)Tj
/F11 1 Tf
0.4804 0 TD
(|)Tj
/F10 1 Tf
0.4582 0 TD
(n)Tj
/F8 1 Tf
10.9091 0 0 10.9091 317.5303 443.6334 Tm
(h)Tj
/F2 1 Tf
0.5762 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4167 0 TD
(\))Tj
ET
317.49 438.81 19.316 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 324.9156 428.7701 Tm
()Tj
/F7 1 Tf
-8.117 -2.9858 TD
()Tj
/F2 1 Tf
1.4527 0.6764 TD
(1)Tj
ET
247.89 399.166 14.151 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 247.8655 388.7133 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 258.0593 386.936 Tm
()Tj
/F9 1 Tf
10.9091 0 0 10.9091 263.2119 408.3058 Tm
( )Tj
/F8 1 Tf
0.5973 -1.11 TD
(N)Tj
/F7 1 Tf
1.1347 0 TD
()Tj
/F8 1 Tf
1.1096 0.6764 TD
(N)Tj
ET
294.21 399.166 9.955 -0.436 re
f
BT
10.9091 0 0 10.9091 296.9146 388.7133 Tm
()Tj
/F2 1 Tf
0.9964 0.686 TD
0.2222 Tc
(+1)Tj
/F9 1 Tf
1.5 1.11 TD
0 Tc
(\012)Tj
/F2 1 Tf
0.8195 -1.11 TD
(+)Tj
/F8 1 Tf
1.1096 0.6764 TD
(h)Tj
/F2 1 Tf
0.5762 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4167 0 TD
(\))Tj
ET
345.21 399.166 19.316 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 352.5779 388.7133 Tm
()Tj
1.3128 1.3625 TD
(N)Tj
ET
366.93 399.166 9.955 -0.436 re
f
BT
10.9091 0 0 10.9091 369.604 388.7133 Tm
()Tj
/F7 1 Tf
1.0519 0.686 TD
()Tj
/F8 1 Tf
1.1652 0.6764 TD
(N)Tj
/F2 1 Tf
1.1347 0 TD
0.2222 Tc
(+2)Tj
ET
393.81 399.166 28.742 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 405.8879 388.7133 Tm
0 Tc
()Tj
1.6353 0.686 TD
(.)Tj
/F2 1 Tf
-27.9518 -2.2753 TD
[(Com)27.8(bining)-333.3(this)-333.2(\(taking)]TJ
/F8 1 Tf
10.5056 0 TD
(N)Tj
/F2 1 Tf
1.1902 0 TD
(=)Tj
/F8 1 Tf
1.0556 0 TD
(x/q)Tj
/F2 1 Tf
1.5538 0 TD
[(\))-333.3(with)-333.3(the)-333.4(previous)-333.2(displa)27.8(y)-333.3(yields)]TJ
/F9 1 Tf
-12.0457 -1.3714 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 138.932 332.2551 Tm
(n)Tj
/F15 1 Tf
0.6446 0 TD
()Tj
/F14 1 Tf
0.8265 0 TD
(x/q)Tj
/F7 1 Tf
10.9091 0 0 10.9091 165.5422 346.0504 Tm
(A)Tj
/F2 1 Tf
0.7985 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(;)Tj
/F8 1 Tf
0.4445 0 TD
[(,)-166.6(nq)]TJ
/F2 1 Tf
1.9437 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F8 1 Tf
1.1651 0.6765 TD
(h)Tj
/F2 1 Tf
0.5762 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
ET
230.73 349.006 20.032 -0.436 re
f
BT
10.9091 0 0 10.9091 235.4232 338.5674 Tm
(2)Tj
/F8 1 Tf
0.5 0 TD
(q)Tj
/F7 1 Tf
1.0189 0.6859 TD
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5716 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F2 1 Tf
1.1651 0.6765 TD
(3\()Tj
/F8 1 Tf
0.8889 0 TD
(x)Tj
/F2 1 Tf
0.7938 0 TD
0.2222 Tc
(+2)Tj
/F8 1 Tf
1.5 0 TD
0 Tc
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
ET
291.21 349.006 44.223 -0.436 re
f
BT
10.9091 0 0 10.9091 305.6448 338.5674 Tm
(2)Tj
/F8 1 Tf
0.5 0 TD
0.0359 Tc
(q)Tj
/F7 1 Tf
2.336 0.6859 TD
0 Tc
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F2 1 Tf
1.1651 0.6765 TD
(3)Tj
ET
375.81 349.006 5.455 -0.436 re
f
BT
10.9091 0 0 10.9091 375.7539 338.5674 Tm
(2)Tj
/F9 1 Tf
1.1904 1.636 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 384.2222 332.2551 Tm
(n)Tj
/F15 1 Tf
0.6447 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(x/q)Tj
/F8 1 Tf
10.9091 0 0 10.9091 412.0289 353.4306 Tm
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 417.3699 351.6533 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 421.327 353.4306 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(nq)Tj
/F2 1 Tf
1.0826 0 TD
(\))Tj
ET
412.05 348.57 29.593 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 419.7499 338.5674 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 429.9427 336.79 Tm
()Tj
/F7 1 Tf
10.9091 0 0 10.9091 442.8164 346.0504 Tm
(A)Tj
/F2 1 Tf
0.7985 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(,)Tj
/F2 1 Tf
-31.8493 -2.9421 TD
[(whic)27.8(h)-333.3(implies)-333.2(the)-333.4(prop)-27.7(osition)-333.3(with)]TJ
/F8 1 Tf
15.5139 0 TD
(b)Tj
/F2 1 Tf
0.7069 0 TD
(=)Tj
/F8 1 Tf
1.0556 0 TD
(nq)Tj
/F2 1 Tf
1.4158 0 TD
[(for)-333.3(some)]TJ
/F8 1 Tf
4.0362 0 TD
(n)Tj
/F7 1 Tf
0.878 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(x/q)Tj
/F2 1 Tf
1.5539 0 TD
(.)Tj
-24.2067 -2.1826 TD
[(W)83.3(e)-258.1(t)27.8(ypically)-258.1(apply)-258.1(this)-258.2(theorem)-258(with)]TJ
/F8 1 Tf
16.2239 0 TD
(h)Tj
/F2 1 Tf
0.5761 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
/F8 1 Tf
0.6667 0 TD
(>)Tj
/F2 1 Tf
1.0555 0 TD
(log)Tj
/F14 1 Tf
7.9701 0 0 7.9701 366.3731 294.5644 Tm
(A)Tj
/F8 1 Tf
10.9091 0 0 10.9091 375.0327 290.144 Tm
(x)Tj
/F2 1 Tf
0.8297 0 TD
[(for)-258.1(some)-258(large)]TJ
/F8 1 Tf
6.2577 0 TD
(A)Tj
/F2 1 Tf
0.75 0 TD
[(.)-419.3(This)]TJ
-31.3255 -1.2694 TD
[(is)-269.4(easily)-269.4(organized)-269.4(if,)-282.2(sa)27.8(y)83.3(,)]TJ
/F8 1 Tf
11.0866 0 TD
(h)Tj
/F2 1 Tf
0.5762 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
0.0944 Tc
(1+)Tj
/F8 1 Tf
1.4667 0 TD
0 Tc
( )Tj
/F2 1 Tf
0.8018 0 TD
(for)Tj
/F7 1 Tf
1.4667 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
0.0359 Tc
[( z)-8.1(/)]TJ
/F2 1 Tf
1.7081 0 TD
0 Tc
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F2 1 Tf
0.7785 0 TD
(primes)Tj
/F8 1 Tf
3.1666 0 TD
(p)Tj
/F7 1 Tf
0.7809 0 TD
()Tj
/F2 1 Tf
0.9445 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
0.044 Tc
(z/)Tj
/F2 1 Tf
1.009 0 TD
0 Tc
(2)Tj
/F8 1 Tf
0.5 0 TD
0.1667 Tc
(,z)Tj
/F2 1 Tf
0.9535 0 TD
0 Tc
[(\))-269.4(where)]TJ
/F8 1 Tf
-30.756 -1.2694 TD
(z)Tj
/F7 1 Tf
0.7868 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
[(,)-312.6(and)-307.4(b)27.8(y)-307.4(letting)]TJ
/F8 1 Tf
7.0127 0 TD
(q)Tj
/F2 1 Tf
0.7897 0 TD
[(b)-27.8(e)-307.4(the)-307.4(pro)-27.7(duct)-307.4(of)-307.4([)]TJ
/F8 1 Tf
8.1492 0 TD
0.0359 Tc
[( z)-8.1(/)]TJ
/F2 1 Tf
1.7081 0 TD
0 Tc
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.5089 0 TD
[(])-307.4(o)0(f)-307.5(these)-307.3(primes)-307.4(so)-307.5(that)]TJ
/F8 1 Tf
-23.4993 -1.2694 TD
(q)Tj
/F2 1 Tf
0.7601 0 TD
(=)Tj
/F8 1 Tf
1.0555 0 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 143.6862 252.5579 Tm
0.0358 Tc
(z)Tj
/F1 1 Tf
1.1055 0 TD
0 Tc
(\(1+)Tj
/F14 1 Tf
1.7709 0 TD
(o)Tj
/F1 1 Tf
0.5135 0 TD
(\(1\)\))Tj
/F2 1 Tf
10.9091 0 0 10.9091 185.3161 248.5998 Tm
[(.)-414.6(W)83.3(e)-243.8(can)-243.7(select)-243.7(an)27.8(y)]TJ
/F8 1 Tf
8.5063 0 TD
()Tj
/F2 1 Tf
0.6604 0 TD
[(in)-243.7(the)-243.8(range)]TJ
/F8 1 Tf
5.3452 0 TD
(x)Tj
/F7 1 Tf
0.8492 0 TD
()Tj
/F8 1 Tf
1.0556 0 TD
()Tj
/F7 1 Tf
0.6944 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(x/)Tj
/F2 1 Tf
1.2382 0 TD
(exp\(\()Tj
/F8 1 Tf
2.3056 0 TD
( )Tj
/F1 1 Tf
7.9701 0 0 7.9701 427.9651 252.5579 Tm
(2)Tj
/F8 1 Tf
10.9091 0 0 10.9091 432.6974 248.5998 Tm
(/)Tj
/F2 1 Tf
0.5 0 TD
(2\))Tj
/F8 1 Tf
0.889 0 TD
0.0439 Tc
(z/)Tj
/F2 1 Tf
1.1756 0 TD
0 Tc
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\).)Tj
-31.2968 -1.2694 TD
[(Prop)-27.7(osition)-446.2(2.1)-446.3(allo)27.8(ws)-446.3(us)-446.2(to)-446.3(handle)-446.2(sequences)-446.2(for)-446.2(whic)27.8(h)]TJ
/F8 1 Tf
25.6471 0 TD
(h)Tj
/F2 1 Tf
0.5762 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
[(\))-446.3(i)0(s)-446.3(signi-)]TJ
-29.1245 -1.2694 TD
[(can)27.8(tly)-307(larger)-307.1(than)-307(1)-307(for)-307(man)27.8(y)-307.1(primes.)-435.6(Therefore)-306.9(w)27.8(e)-307.1(will,)-312.3(from)-307(no)27.8(w)-307.1(on,)-312.3(restrict)]TJ
T*
[(ourselv)27.8(es)-299.5(to)-299.5(the)-299.5(case)-299.4(when)-299.5(0)]TJ
/F7 1 Tf
12.5169 0 TD
()Tj
/F8 1 Tf
1.0556 0 TD
(h)Tj
/F2 1 Tf
0.5761 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(n)Tj
/F2 1 Tf
0.6002 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F2 1 Tf
1.0556 0 TD
[(1)-299.5(for)-299.5(all)]TJ
/F8 1 Tf
3.6512 0 TD
(n)Tj
/F2 1 Tf
0.6002 0 TD
[(.)-433.2(Supp)-27.8(ose)-299.4(that)-299.5(\()]TJ
/F8 1 Tf
7.1211 0 TD
0.0359 Tc
(q,)Tj
/F7 1 Tf
0.9268 0 TD
0 Tc
(S)Tj
/F2 1 Tf
0.6806 0 TD
[(\))-277.8(=)-277.7(1)-299.4(and)]TJ
-29.8398 -1.2694 TD
[(dene)-333.3()]TJ
/F14 1 Tf
7.9701 0 0 7.9701 159.4052 191.5706 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 167.0007 193.2074 Tm
0.2777 Tc
(=)Tj
/F14 1 Tf
7.9701 0 0 7.9701 187.6065 191.5706 Tm
0 Tc
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 192.1714 193.2074 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
0.3333 Tc
[(\)b)361.1(y)]TJ
-3.6774 -2.387 TD
0 Tc
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 171.6234 165.5313 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 176.1883 167.1671 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
[(\))-277.8(:)0(=)-981.1(max)]TJ
/F14 1 Tf
7.9701 0 0 7.9701 208.4832 159.3713 Tm
(x/)Tj
/F1 1 Tf
1.1294 0 TD
(4)Tj
/F15 1 Tf
0.5313 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(y)Tj
/F15 1 Tf
0.5611 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(x)Tj
/F2 1 Tf
10.9091 0 0 10.9091 260.7701 167.1671 Tm
(max)Tj
/F14 1 Tf
7.9701 0 0 7.9701 249.5851 159.3713 Tm
(a)Tj
/F1 1 Tf
1.6269 0 TD
0.0001 Tc
[(\(mo)-29.3(d)]TJ
/F14 1 Tf
2.8039 0 TD
0 Tc
(q)Tj
/F1 1 Tf
0.5103 0 TD
(\))Tj
/F9 1 Tf
10.9091 0 0 10.9091 297.7126 179.7131 Tm
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F7 1 Tf
0.3333 0.05 TD
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(y)Tj
/F2 1 Tf
0.5261 0 TD
(;)Tj
/F8 1 Tf
0.4445 0 TD
0.0359 Tc
[(q,)-130.8(a)]TJ
/F2 1 Tf
1.4553 0 TD
0 Tc
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1.1096 0.6821 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 364.8787 172.9713 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 369.4436 174.6082 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(a)Tj
/F2 1 Tf
0.5286 0 TD
(\))Tj
ET
359.49 170.086 24.157 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 363.8795 159.6842 Tm
0.0359 Tc
(q)Tj
/F14 1 Tf
7.9701 0 0 7.9701 374.7886 158.0482 Tm
0 Tc
(q)Tj
/F8 1 Tf
10.9091 0 0 10.9091 386.3333 174.5474 Tm
(y)Tj
ET
386.13 170.086 6.235 -0.436 re
f
BT
10.9091 0 0 10.9091 386.0862 159.6842 Tm
(x)Tj
/F7 1 Tf
0.6811 0.6859 TD
(A)Tj
/F2 1 Tf
0.7985 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F9 1 Tf
0.3889 1.15 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
0.6667 1.46 TD
()Tj
/F7 1 Tf
1.4874 -0.7335 TD
(A)Tj
/F2 1 Tf
0.7985 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
ET
440.49 170.086 23.43 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 442.0374 159.6842 Tm
()Tj
/F2 1 Tf
0.5958 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
/F8 1 Tf
0.6444 0.6859 TD
(.)Tj
/F2 1 Tf
-31.7416 0 TD
(\(2.4\))Tj
0 -2.3921 TD
[(In)-271(view)-270.8(of)-271(\(2.2\))-271(it)-270.9(seems)-270.8(more)-271(natural)-270.8(to)-271(consider)]TJ
/F7 1 Tf
21.429 0 TD
(|A)Tj
/F2 1 Tf
1.0763 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(y)Tj
/F2 1 Tf
0.5261 0 TD
(;)Tj
/F8 1 Tf
0.4445 0 TD
0.0359 Tc
[(q,)-130.7(a)]TJ
/F2 1 Tf
1.4553 0 TD
0 Tc
(\))Tj
/F7 1 Tf
0.4863 0 TD
()Tj
/F8 1 Tf
0.8753 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 415.214 139.4351 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 419.7788 141.071 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(a)Tj
/F2 1 Tf
0.5286 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
0.0359 Tc
(q)Tj
/F14 1 Tf
7.9701 0 0 7.9701 454.6371 139.4351 Tm
0 Tc
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 459.202 141.071 Tm
(\))Tj
/F7 1 Tf
0.3889 0 TD
(A)Tj
/F2 1 Tf
0.7985 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(y)Tj
/F2 1 Tf
0.5261 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F2 1 Tf
-33.6948 -1.2694 TD
[(instead)-324.5(of)-324.6(\(2.4\))-324.5(ab)-27.8(o)27.8(v)27.8(e)0(.)-441.5(H)0(o)27.8(w)27.8(e)0(v)27.8(e)0(r)-324.5(\(2.4\))-324.6(seems)-324.5(to)-324.5(b)-27.8(e)-324.5(the)-324.6(most)-324.6(con)27.9(v)27.8(enien)27.8(t)-324.5(w)27.7(a)27.9(y)-324.6(to)]TJ
ET
endstream
endobj
52 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F10 13 0 R
/F11 14 0 R
/F14 22 0 R
/F15 23 0 R
/F16 24 0 R
/F17 28 0 R
/F18 53 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
57 0 obj
<<
/Length 16384
>>
stream
BT
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
(12)Tj
/F5 1 Tf
7.9701 0 0 7.9701 202.119 726.9739 Tm
[(ANDREW)-377.5(GRANVILLE)-377.4(AND)-377.7(K)0(.)-377.7(SOUND)29.4(ARARAJAN)]TJ
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 701.071 Tm
[(form)27.9(ulate)-334.1(our)-334.1(results,)-334.2(and)-334.1(should)-334(b)-27.8(e)-334.1(though)27.9(t)-334.1(o)0(f)-334.1(a)0(s)-334.1(incorp)-27.7(orating)-334(a)-334.2(h)27.8(yp)-27.8(othesis)]TJ
0 -1.2694 TD
(that)Tj
/F7 1 Tf
2.2798 0 TD
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(y)Tj
/F2 1 Tf
0.5261 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/y)Tj
/F2 1 Tf
1.4727 0 TD
[(is)-446.4(v)27.8(ery)-446.5(close)-446.4(to)]TJ
/F7 1 Tf
7.272 0 TD
(A)Tj
/F2 1 Tf
0.7985 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5716 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/x)Tj
/F2 1 Tf
1.518 0 TD
(when)Tj
/F8 1 Tf
2.7243 0 TD
(x/)Tj
/F2 1 Tf
1.0715 0 TD
(4)Tj
/F7 1 Tf
0.9663 0 TD
()Tj
/F8 1 Tf
1.2441 0 TD
(y)Tj
/F7 1 Tf
0.9925 0 TD
()Tj
/F8 1 Tf
1.2441 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
[(.)-783.9(F)83.4(ormally)-446.4(w)27.8(e)-446.5(sa)27.8(y)]TJ
-25.6069 -1.2694 TD
(that)Tj
/F7 1 Tf
2.2265 0 TD
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/x)Tj
/F2 1 Tf
1.4646 0 TD
[(is)-393.2(slo)27.9(wly)-393.1(v)55.5(arying:)-563.9(a)-393.1(t)27.7(ypical)-393(case)-393.1(is)-393.1(when)]TJ
/F7 1 Tf
18.3407 0 TD
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/x)Tj
/F2 1 Tf
1.4647 0 TD
[(b)-27.7(eha)27.8(v)27.8(e)0(s)-393.2(lik)27.8(e)-393.1(a)]TJ
-27.792 -1.2694 TD
[(p)-27.8(o)27.9(w)27.7(e)0(r)-307.7(o)0(f)-307.9(log)]TJ
/F8 1 Tf
5.4656 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
[(,)-312.9(a)-307.9(feature)-307.7(seen)-307.7(in)-307.8(the)-307.8(motiv)55.6(ating)-307.8(examples)-307.7(of)]TJ
/F7 1 Tf
20.1226 0 TD
(A)Tj
/F2 1 Tf
1.1062 0 TD
[(b)-27.8(eing)-307.8(the)-307.8(set)-307.7(of)]TJ
-27.2659 -1.2694 TD
[(primes,)-326.1(or)-324.5(sums)-324.4(of)-324.4(t)27.7(w)27.8(o)-324.4(squares.)-441.4(With)-324.4(these)-324.4(preliminaries)-324.3(in)-324.5(place)-324.4(w)27.8(e)-324.4(can)-324.4(no)27.8(w)]TJ
T*
[(form)27.9(ulate)-333.4(our)-333.3(main)-333.3(principle.)]TJ
/F5 1 Tf
2.0091 -2.1827 TD
[(Pr)22.3(oposition)-348.4(2.2)]TJ
/F2 1 Tf
7.8263 0 TD
(.)Tj
/F16 1 Tf
1.148 0 TD
-0.0511 Tc
[(Le)-51.1(t)]TJ
/F8 1 Tf
1.7006 0 TD
0 Tc
(x)Tj
/F16 1 Tf
0.9037 0 TD
[(b)51.1(e)-332.1(lar)51.1(ge)-332.1(and)-332.1(let)]TJ
/F7 1 Tf
6.8869 0 TD
(A)Tj
/F2 1 Tf
0.7985 0 TD
(,)Tj
/F7 1 Tf
0.6151 0 TD
(S)Tj
/F8 1 Tf
1.0127 0 TD
(h)Tj
/F16 1 Tf
0.5762 0 TD
(,)Tj
/F8 1 Tf
0.6439 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 387.2777 606.3841 Tm
(q)Tj
/F16 1 Tf
10.9091 0 0 10.9091 395.466 608.02 Tm
(and)Tj
/F2 1 Tf
1.9166 0 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 425.4655 606.3841 Tm
(q)Tj
/F16 1 Tf
10.9091 0 0 10.9091 433.6538 608.02 Tm
[(b)51.1(e)-332.2(as)-332.1(ab)51.1(ove.)]TJ
-28.8616 -1.2694 TD
-0.0511 Tc
[(Le)-51.1(t)]TJ
/F8 1 Tf
1.7786 0 TD
0 Tc
(q)Tj
/F7 1 Tf
0.8553 0 TD
()Tj
1.1508 0.7203 TD
()Tj
ET
169.17 602.446 6.235 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 169.1786 594.1719 Tm
(x)Tj
/F7 1 Tf
0.9445 0 TD
()Tj
/F8 1 Tf
1.1508 0 TD
()Tj
/F7 1 Tf
0.7897 0 TD
()Tj
/F8 1 Tf
1.1508 0 TD
(x/)Tj
/F2 1 Tf
1.0715 0 TD
(4)Tj
/F16 1 Tf
0.9103 0 TD
[(b)51(e)-410.3(p)51.1(ositive)-410.3(c)51.1(oprime)-410.2(inte)51.2(gers)-410.3(with)]TJ
/F2 1 Tf
14.7783 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
0.0359 Tc
(q,)Tj
/F7 1 Tf
0.9267 0 TD
0 Tc
(S)Tj
/F2 1 Tf
0.6805 0 TD
0.373 Tc
(\)=\()Tj
/F8 1 Tf
2.3016 0 TD
0 Tc
(,)Tj
/F7 1 Tf
0.8612 0 TD
(S)Tj
/F2 1 Tf
0.6805 0 TD
0.373 Tc
(\)=1)Tj
/F16 1 Tf
2.4126 0 TD
0 Tc
(.)Tj
-33.6659 -1.2694 TD
(Then)Tj
/F8 1 Tf
4.675 -1.8879 TD
(q)Tj
ET
162.33 555.286 20.246 -0.436 re
f
BT
10.9091 0 0 10.9091 162.3073 544.8657 Tm
()Tj
/F2 1 Tf
0.5958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
0.4984 0.6859 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 192.8398 550.7128 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 197.4047 552.3487 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5716 0 TD
0.2222 Tc
(\)+)Tj
/F8 1 Tf
2.4075 0.6765 TD
0 Tc
()Tj
ET
226.65 555.286 19.531 -0.436 re
f
BT
10.9091 0 0 10.9091 226.6531 544.8657 Tm
()Tj
/F2 1 Tf
0.5959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4166 0 TD
(\))Tj
0.4985 0.6859 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 256.4702 550.5714 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 260.4274 552.3487 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5716 0 TD
0.2222 Tc
(\)+)Tj
/F8 1 Tf
1.6111 0 TD
0 Tc
(x)Tj
/F15 1 Tf
7.9701 0 0 7.9701 294.7148 556.8528 Tm
()Tj
/F12 1 Tf
4.9813 0 0 4.9813 302.4977 560.1116 Tm
(1)Tj
ET
302.49 559.063 3.39 -0.373 re
f
BT
4.9813 0 0 4.9813 302.4977 553.7275 Tm
(8)Tj
/F7 1 Tf
10.9091 0 0 10.9091 310.6112 552.3487 Tm
()Tj
/F9 1 Tf
1.2778 1.15 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F2 1 Tf
1.4647 0.7265 TD
(1)Tj
ET
329.37 555.286 27.75 -0.436 re
f
BT
10.9091 0 0 10.9091 329.3828 544.8657 Tm
([)Tj
/F8 1 Tf
0.2777 0 TD
(x/)Tj
/F2 1 Tf
1.0716 0 TD
(2)Tj
/F8 1 Tf
0.5 0 TD
()Tj
/F2 1 Tf
0.4166 0 TD
(])Tj
/F9 1 Tf
1.3803 1.636 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 360.1465 538.5535 Tm
(s)Tj
/F15 1 Tf
0.4913 0 TD
()Tj
/F14 1 Tf
0.8265 0 TD
(x/)Tj
/F1 1 Tf
1.1292 0 TD
(\(2)Tj
/F14 1 Tf
0.9445 0 TD
()Tj
/F1 1 Tf
0.434 0 TD
(\))Tj
/F8 1 Tf
10.9091 0 0 10.9091 396.9435 559.7898 Tm
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 402.2845 558.153 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 406.8493 559.7898 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(s)Tj
/F2 1 Tf
0.4688 0 TD
(\))Tj
ET
396.93 555.286 23.504 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 403.5896 544.8657 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 409.2374 543.2299 Tm
(q)Tj
/F7 1 Tf
10.9091 0 0 10.9091 424.0677 552.3487 Tm
()Tj
/F2 1 Tf
1 0 TD
(1)Tj
/F9 1 Tf
0.5 1.15 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F8 1 Tf
0.3333 0.05 TD
(.)Tj
/F16 1 Tf
-27.8071 -4.3931 TD
[(Pr)51.2(o)51.1(o)0(f)]TJ
/F2 1 Tf
2.3267 0 TD
[(.)-1191.4(Let)]TJ
/F8 1 Tf
3.3113 0 TD
(R)Tj
/F2 1 Tf
1.1288 0 TD
[(:=)-361.8([)]TJ
/F8 1 Tf
1.6952 0 TD
(x/)Tj
/F2 1 Tf
1.0715 0 TD
(\(4)Tj
/F8 1 Tf
0.889 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\)])Tj
/F7 1 Tf
1.0285 0 TD
()Tj
1.1395 0.7203 TD
()Tj
ET
292.41 512.686 6.235 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 292.4194 504.4244 Tm
(x/)Tj
/F2 1 Tf
1.0715 0 TD
[(5)-383.7(and)]TJ
/F8 1 Tf
2.8786 0 TD
(S)Tj
/F2 1 Tf
1.0327 0 TD
[(:=)-361.8([)]TJ
/F8 1 Tf
1.6952 0 TD
(x/)Tj
/F2 1 Tf
1.0715 0 TD
(\(2)Tj
/F8 1 Tf
0.8889 0 TD
()Tj
/F2 1 Tf
0.4167 0 TD
(\)])Tj
/F7 1 Tf
1.0285 0 TD
()Tj
1.1395 0.7203 TD
()Tj
ET
423.93 512.686 6.235 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 423.9442 504.4244 Tm
(x/)Tj
/F2 1 Tf
1.0715 0 TD
[(2.)-595.7(W)83.3(e)-383.8(sum)]TJ
-29.0431 -1.2694 TD
[(the)-347.4(v)55.5(alues)-347.5(of)]TJ
/F8 1 Tf
5.8813 0 TD
(a)Tj
/F2 1 Tf
0.5286 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(n)Tj
/F2 1 Tf
0.6002 0 TD
0.3475 Tc
[(\)a)347.5(s)]TJ
/F8 1 Tf
1.9784 0 TD
0 Tc
(n)Tj
/F2 1 Tf
0.9477 0 TD
[(v)55.5(aries)-347.4(o)27.8(v)27.8(e)0(r)-347.4(the)-347.4(in)27.8(tegers)-347.4(in)-347.5(the)-347.4(follo)27.8(wing)]TJ
/F8 1 Tf
17.5906 0 TD
(R)Tj
/F7 1 Tf
0.9987 0 TD
()Tj
/F8 1 Tf
1.0094 0 TD
(S)Tj
/F2 1 Tf
1.0183 0 TD
(Maier)Tj
-30.942 -1.2694 TD
(matrix.)Tj
ET
154.77 464.409 298.688 -0.398 re
f
154.77 464.03 0.399 -120.5 re
f
BT
10.9091 0 0 10.9091 158.1479 451.2946 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(R)Tj
/F2 1 Tf
0.9892 0 TD
0.2222 Tc
[(+1)222.2(\))]TJ
/F8 1 Tf
1.889 0 TD
0 Tc
(q)Tj
/F2 1 Tf
0.7045 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
1.33 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(R)Tj
/F2 1 Tf
0.9892 0 TD
0.2222 Tc
[(+1)222.2(\))]TJ
/F8 1 Tf
1.8889 0 TD
0 Tc
(q)Tj
/F2 1 Tf
0.7046 0 TD
0.2222 Tc
(+2)Tj
/F8 1 Tf
1.5 0 TD
0 Tc
()Tj
/F7 1 Tf
4.005 0 TD
0.1667 Tc
()Tj
/F2 1 Tf
4.9217 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
(R)Tj
/F2 1 Tf
0.9892 0 TD
0.2222 Tc
[(+1)222.2(\))]TJ
/F8 1 Tf
1.889 0 TD
0 Tc
(q)Tj
/F2 1 Tf
0.7045 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
0.0577 Tc
(S)Tj
/F2 1 Tf
-25.6712 -1.2694 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
(R)Tj
/F2 1 Tf
0.9892 0 TD
0.2222 Tc
[(+2)222.2(\))]TJ
/F8 1 Tf
1.889 0 TD
0 Tc
(q)Tj
/F2 1 Tf
0.7045 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
1.33 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(R)Tj
/F2 1 Tf
0.9892 0 TD
0.2222 Tc
[(+2)222.2(\))]TJ
/F8 1 Tf
1.8889 0 TD
0 Tc
(q)Tj
/F2 1 Tf
0.7046 0 TD
0.2222 Tc
(+2)Tj
/F8 1 Tf
1.5 0 TD
0 Tc
()Tj
/F7 1 Tf
4.005 0 TD
0.1667 Tc
()Tj
/F2 1 Tf
4.9217 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
(R)Tj
/F2 1 Tf
0.9892 0 TD
0.2222 Tc
[(+2)222.2(\))]TJ
/F8 1 Tf
1.889 0 TD
0 Tc
(q)Tj
/F2 1 Tf
0.7045 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
0.0577 Tc
(S)Tj
/F2 1 Tf
-25.6712 -1.7649 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
(R)Tj
/F2 1 Tf
0.9892 0 TD
0.2222 Tc
[(+3)222.2(\))]TJ
/F8 1 Tf
1.889 0 TD
0 Tc
(q)Tj
/F2 1 Tf
0.7045 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
()Tj
/F7 1 Tf
4.1352 0 TD
6.9213 Tc
()Tj
/F2 1 Tf
14.4837 0.7306 TD
0 Tc
(.)Tj
0 -0.3653 TD
(.)Tj
T*
(.)Tj
-23.5903 -2.2107 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(R)Tj
/F2 1 Tf
0.9892 0 TD
0.2222 Tc
[(+4)222.2(\))]TJ
/F8 1 Tf
1.889 0 TD
0 Tc
(q)Tj
/F2 1 Tf
0.7045 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
4.1352 0.7306 TD
(.)Tj
0 -0.3653 TD
(.)Tj
T*
(.)Tj
ET
301.05 414.009 72.912 -0.399 re
f
301.05 413.684 0.399 -33.674 re
f
BT
10.9091 0 0 10.9091 304.4752 400.9574 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
-0.0278 Tc
[(r,)-194.4(s)]TJ
/F2 1 Tf
1.3365 0 TD
0 Tc
(\))Tj
/F1 1 Tf
7.9701 0 0 7.9701 327.5417 404.9165 Tm
(th)Tj
/F2 1 Tf
10.9091 0 0 10.9091 339.6732 400.9574 Tm
[(en)27.8(try)-277.8(:)]TJ
-3.1131 -1.2694 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(R)Tj
/F2 1 Tf
0.9892 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
(r)Tj
/F2 1 Tf
0.4789 0 TD
(\))Tj
/F8 1 Tf
0.389 0 TD
(q)Tj
/F2 1 Tf
0.7045 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
(s)Tj
ET
373.65 413.684 0.399 -33.674 re
f
301.05 379.929 72.912 -0.398 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 415.4969 402.0463 Tm
(.)Tj
0 -0.3653 TD
(.)Tj
T*
(.)Tj
-21.0351 -1.9835 TD
(.)Tj
0 -0.3653 TD
(.)Tj
T*
(.)Tj
6.5514 0.7306 TD
(.)Tj
0 -0.3653 TD
(.)Tj
T*
(.)Tj
/F7 1 Tf
7.0324 0 TD
()Tj
/F2 1 Tf
7.4512 0.7306 TD
(.)Tj
0 -0.3653 TD
(.)Tj
T*
(.)Tj
-22.5903 -1.2694 TD
(2)Tj
/F8 1 Tf
0.5 0 TD
0.0077 Tc
(Rq)Tj
/F2 1 Tf
1.4715 0 TD
0 Tc
(+)Tj
/F8 1 Tf
1 0 TD
()Tj
/F7 1 Tf
4.6074 0 TD
0.1666 Tc
[()-5865.8()-0.1()]TJ
/F2 1 Tf
13.1208 0 TD
0 Tc
(2)Tj
/F8 1 Tf
0.5 0 TD
0.0077 Tc
(Rq)Tj
/F2 1 Tf
1.4715 0 TD
0 Tc
(+)Tj
/F8 1 Tf
1 0 TD
0.0577 Tc
(S)Tj
ET
453.09 464.03 0.398 -120.5 re
f
154.77 343.449 298.688 -0.399 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 322.5445 Tm
0 Tc
[(W)83.3(e)-333.5(sum)-333.4(the)-333.4(v)55.5(alues)-333.5(of)]TJ
/F8 1 Tf
9.6784 0 TD
(a)Tj
/F2 1 Tf
0.5286 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(n)Tj
/F2 1 Tf
0.6003 0 TD
[(\))-333.4(i)0(n)-333.5(t)27.8(w)27.8(o)-333.5(w)27.8(a)27.8(ys:)-444.6(rst)-333.4(ro)27.8(w)-333.4(b)27.8(y)-333.5(r)0(o)27.8(w)0(,)-333.4(and)-333.4(second,)-333.4(column)]TJ
-11.1962 -1.2694 TD
[(b)27.8(y)-430.7(column.)-736.1(Note)-430.6(that)-430.5(the)]TJ
/F8 1 Tf
12.2643 0 TD
(n)Tj
/F2 1 Tf
1.0309 0 TD
[(app)-27.7(earing)-430.6(in)-430.6(our)-430.6(matrix)-430.6(all)-430.5(lie)-430.6(b)-27.8(e)0(t)27.8(w)27.8(een)]TJ
/F8 1 Tf
19.1059 0 TD
(x/)Tj
/F2 1 Tf
1.0715 0 TD
(4)Tj
-33.4726 -1.2694 TD
(and)Tj
/F8 1 Tf
1.9445 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(.)Tj
-0.5068 -1.2694 TD
(The)Tj
/F8 1 Tf
2.2553 0 TD
(r)Tj
/F1 1 Tf
7.9701 0 0 7.9701 170.5455 284.9594 Tm
(th)Tj
/F2 1 Tf
10.9091 0 0 10.9091 184.8578 281.0002 Tm
[(ro)27.8(w)-533.1(con)27.9(tributes)]TJ
/F7 1 Tf
7.5218 0 TD
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\(\()Tj
/F8 1 Tf
0.7778 0 TD
(R)Tj
/F2 1 Tf
1.1224 0 TD
(+)Tj
/F8 1 Tf
1.1332 0 TD
(r)Tj
/F2 1 Tf
0.479 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.8377 0 TD
(+)Tj
/F8 1 Tf
1.1332 0 TD
(S)Tj
/F2 1 Tf
1.0875 0 TD
(;)Tj
/F8 1 Tf
0.4445 0 TD
(,)Tj
/F2 1 Tf
0.8611 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(R)Tj
/F2 1 Tf
1.1224 0 TD
(+)Tj
/F8 1 Tf
1.1332 0 TD
(r)Tj
/F2 1 Tf
0.4789 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
/F7 1 Tf
0.7443 0 TD
0.3555 Tc
(A)Tj
/F2 1 Tf
1.9317 0 TD
0 Tc
(\(\()Tj
/F8 1 Tf
0.7778 0 TD
(R)Tj
/F2 1 Tf
1.1224 0 TD
(+)Tj
/F8 1 Tf
1.1332 0 TD
(r)Tj
/F2 1 Tf
0.4789 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(;)Tj
/F8 1 Tf
-33.6948 -1.2694 TD
(,)Tj
/F2 1 Tf
0.8611 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(R)Tj
/F2 1 Tf
0.8835 0 TD
(+)Tj
/F8 1 Tf
0.8943 0 TD
(r)Tj
/F2 1 Tf
0.4789 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
[(\).)-426.8(Using)-280.4(\(2.4\),)-291(and)-280.5(noting)-280.4(that)]TJ
/F8 1 Tf
13.5398 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 319.6065 265.3749 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 323.5646 267.1522 Tm
(\(\()Tj
/F8 1 Tf
0.7777 0 TD
(R)Tj
/F2 1 Tf
0.8836 0 TD
(+)Tj
/F8 1 Tf
0.8942 0 TD
(r)Tj
/F2 1 Tf
0.479 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
0.2778 Tc
(\)=)Tj
/F8 1 Tf
1.7222 0 TD
0 Tc
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 390.3004 265.3749 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 394.2586 267.1522 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(R)Tj
/F2 1 Tf
0.8836 0 TD
(+)Tj
/F8 1 Tf
0.8942 0 TD
(r)Tj
/F2 1 Tf
0.4789 0 TD
0.2805 Tc
[(\)a)280.5(s\()]TJ
/F8 1 Tf
2.2332 0 TD
0 Tc
[(,)-166.6(q)]TJ
/F2 1 Tf
1.3434 0 TD
0.2778 Tc
[(\)=1)277.8(,)]TJ
-31.4726 -1.2694 TD
-0.0278 Tc
[(we)-361.2(h)-27.8(ave)]TJ
/F8 1 Tf
9.3889 -1.1896 TD
0 Tc
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 226.5634 238.5494 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 230.5215 240.3268 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(R)Tj
/F2 1 Tf
0.9892 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
(r)Tj
/F2 1 Tf
0.479 0 TD
(\))Tj
ET
221.25 235.846 44.708 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 236.5021 225.4636 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 246.6949 223.6861 Tm
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 268.3228 240.3268 Tm
(S)Tj
ET
268.29 235.846 11.864 -0.436 re
f
BT
10.9091 0 0 10.9091 271.1372 225.4636 Tm
(x)Tj
/F7 1 Tf
0.9391 0.6859 TD
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
0.2222 Tc
(\)+)Tj
/F8 1 Tf
1.6111 0 TD
0 Tc
(O)Tj
/F9 1 Tf
0.7906 1.11 TD
( )Tj
/F2 1 Tf
1.0039 -0.4335 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 346.8124 238.5494 Tm
()Tj
ET
1 i
334.53 235.41 19.531 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 334.4807 225.4636 Tm
()Tj
/F2 1 Tf
0.5958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4167 0 TD
(\))Tj
/F7 1 Tf
0.4984 0.6859 TD
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F9 1 Tf
0.3889 1.11 TD
(\012)Tj
/F8 1 Tf
0.5973 -1.11 TD
(.)Tj
/F2 1 Tf
-24.4156 -2.7216 TD
[(Summing)-318.6(this)-318.6(o)27.8(v)27.8(e)0(r)-318.7(all)-318.7(the)-318.7(ro)27.8(ws)-318.7(w)27.7(e)-318.7(see)-318.7(that)-318.7(the)-318.7(sum)-318.7(of)]TJ
/F8 1 Tf
24.019 0 TD
(a)Tj
/F14 1 Tf
7.9701 0 0 7.9701 386.5913 201.6199 Tm
(n)Tj
/F2 1 Tf
10.9091 0 0 10.9091 395.7041 203.2567 Tm
(with)Tj
/F8 1 Tf
2.2631 0 TD
(n)Tj
/F2 1 Tf
0.919 0 TD
[(ranging)-318.6(o)27.8(v)27.8(e)0(r)]TJ
-28.565 -1.2694 TD
[(the)-333.3(Maier)-333.3(matrix)-333.3(ab)-27.8(o)27.9(v)27.8(e)-333.4(equals)]TJ
0 -3.1761 TD
(\(2)Tj
/F8 1 Tf
0.8889 0 TD
(.)Tj
/F2 1 Tf
0.2778 0 TD
(5a\))Tj
/F8 1 Tf
7.3882 0.6765 TD
(S)Tj
ET
212.13 157.726 11.863 -0.436 re
f
BT
10.9091 0 0 10.9091 214.94 147.2777 Tm
(x)Tj
/F7 1 Tf
0.9391 0.6859 TD
(A)Tj
/F2 1 Tf
0.7985 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F1 1 Tf
7.9701 0 0 7.9701 259.0486 168.3975 Tm
(2)Tj
/F14 1 Tf
0.5312 0 TD
(R)Tj
/F9 1 Tf
10.9091 0 0 10.9091 256.5001 165.1247 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 250.4339 141.4964 Tm
(r)Tj
/F1 1 Tf
0.5089 0 TD
(=)Tj
/F14 1 Tf
0.8265 0 TD
(R)Tj
/F1 1 Tf
0.8064 0 TD
(+1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 281.338 162.141 Tm
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 286.679 160.3636 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 290.6371 162.141 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(r)Tj
/F2 1 Tf
0.4789 0 TD
(\))Tj
ET
281.37 157.726 23.008 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 285.7664 147.2777 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 295.9602 145.5003 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 307.9661 154.7606 Tm
(+)Tj
/F8 1 Tf
1 0 TD
(O)Tj
/F9 1 Tf
0.7905 1.11 TD
( )Tj
/F2 1 Tf
1.0039 -0.4335 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 347.5417 160.3636 Tm
()Tj
ET
335.25 157.29 19.531 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 335.21 147.2777 Tm
()Tj
/F2 1 Tf
0.5958 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4166 0 TD
(\))Tj
/F7 1 Tf
0.4985 0.6859 TD
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(R)Tj
/F9 1 Tf
0.767 1.11 TD
(\012)Tj
/F8 1 Tf
0.5972 -1.11 TD
(.)Tj
ET
endstream
endobj
58 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F12 15 0 R
/F14 22 0 R
/F15 23 0 R
/F16 24 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
60 0 obj
<<
/Length 19415
>>
stream
BT
/F5 1 Tf
7.9701 0 0 7.9701 162.3741 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
[(AN)-377.7(UNCER)87.7(T)87.5(AINTY)-377.5(PRINCIPLE)-377.6(F)29.2(O)0(R)-377.8(ARITHMETIC)-377.4(SEQUENCES)]TJ
/F2 1 Tf
10.9091 0 0 10.9091 478.5006 726.9741 Tm
(13)Tj
-30.9635 -2.3745 TD
[(The)-317.2(con)27.9(tribution)-317.2(of)-317.2(column)]TJ
/F8 1 Tf
12.3271 0 TD
(s)Tj
/F2 1 Tf
0.7859 0 TD
(is)Tj
/F7 1 Tf
0.9894 0 TD
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\(2)Tj
/F8 1 Tf
0.889 0 TD
0.0077 Tc
(Rq)Tj
/F2 1 Tf
1.4392 0 TD
0 Tc
(+)Tj
/F8 1 Tf
0.9678 0 TD
(s)Tj
/F2 1 Tf
0.8854 0 TD
(;)Tj
/F8 1 Tf
0.4445 0 TD
0.0359 Tc
[(q,)-130.7()35.9(s)]TJ
/F2 1 Tf
1.8121 0 TD
0 Tc
(\))Tj
/F7 1 Tf
0.5788 0 TD
0.1899 Tc
(A)Tj
/F2 1 Tf
1.7661 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.389 0 TD
0.0077 Tc
(Rq)Tj
/F2 1 Tf
1.4392 0 TD
0 Tc
(+)Tj
/F8 1 Tf
0.9678 0 TD
(s)Tj
/F2 1 Tf
0.8854 0 TD
(;)Tj
/F8 1 Tf
0.4445 0 TD
0.0359 Tc
[(q,)-130.8()35.9(s)]TJ
/F2 1 Tf
1.8121 0 TD
0 Tc
[(\).)-439.1(By)]TJ
-31.6308 -1.2694 TD
[(\(2.4\),)-333.3(and)-333.3(since)]TJ
/F8 1 Tf
7.0611 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 201.1706 685.587 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 205.7355 687.2229 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(s)Tj
/F2 1 Tf
0.8854 0 TD
0.2778 Tc
(\)=)Tj
/F8 1 Tf
1.7222 0 TD
0 Tc
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 243.7659 685.587 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 248.3308 687.2229 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(s)Tj
/F2 1 Tf
0.4687 0 TD
0.3334 Tc
[(\)a)333.4(s\()]TJ
/F8 1 Tf
2.3389 0 TD
0 Tc
[(,)-166.6(q)]TJ
/F2 1 Tf
1.3434 0 TD
[(\))-277.8(=)-277.8(1)0(,)-333.3(w)27.7(e)-333.3(see)-333.4(that)-333.3(this)-333.2(is)]TJ
/F8 1 Tf
-6.1667 -1.9796 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 235.9253 663.991 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 240.4902 665.627 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(s)Tj
/F2 1 Tf
0.4687 0 TD
(\))Tj
ET
230.61 661.126 23.504 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 234.5992 650.7029 Tm
0.0359 Tc
(q)Tj
/F14 1 Tf
7.9701 0 0 7.9701 245.5094 649.067 Tm
0 Tc
(q)Tj
/F8 1 Tf
10.9091 0 0 10.9091 256.4802 665.5662 Tm
0.0077 Tc
(Rq)Tj
ET
256.53 661.126 13.629 -0.436 re
f
BT
10.9091 0 0 10.9091 260.1773 650.7029 Tm
0 Tc
(x)Tj
/F7 1 Tf
1.02 0.686 TD
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
0.2223 Tc
(\)+)Tj
/F8 1 Tf
1.6111 0 TD
0 Tc
(O)Tj
/F9 1 Tf
0.7906 1.1099 TD
( )Tj
/F2 1 Tf
1.0089 -0.4279 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 336.7891 663.991 Tm
(q)Tj
ET
324.45 661.126 20.246 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 324.4025 650.7029 Tm
()Tj
/F2 1 Tf
0.5959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
/F7 1 Tf
0.4984 0.686 TD
(A)Tj
/F2 1 Tf
0.7985 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F9 1 Tf
0.389 1.1099 TD
(\012)Tj
/F8 1 Tf
0.5972 -1.11 TD
(.)Tj
/F2 1 Tf
-23.5574 -2.6101 TD
[(Summing)-333.3(this)-333.3(o)27.8(v)27.8(e)0(r)-333.4(all)-333.3(the)-333.3(columns)-333.2(w)27.7(e)-333.3(see)-333.4(that)-333.3(the)-333.3(Maier)-333.3(matrix)-333.3(sum)-333.3(is)]TJ
0 -2.9813 TD
(\(2)Tj
/F8 1 Tf
0.8889 0 TD
(.)Tj
/F2 1 Tf
0.2778 0 TD
(5b\))Tj
/F8 1 Tf
7.856 0.6765 TD
0.0077 Tc
(Rq)Tj
ET
217.17 600.166 13.629 -0.436 re
f
BT
10.9091 0 0 10.9091 220.9255 589.7076 Tm
0 Tc
(x)Tj
/F7 1 Tf
1.02 0.6859 TD
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F14 1 Tf
7.9701 0 0 7.9701 262.3821 610.8274 Tm
(S)Tj
/F9 1 Tf
10.9091 0 0 10.9091 257.3011 607.5547 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 257.8122 584.2361 Tm
(s)Tj
/F1 1 Tf
0.4913 0 TD
(=1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 276.0727 604.6306 Tm
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 281.4137 602.9948 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 285.9786 604.6306 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(s)Tj
/F2 1 Tf
0.4688 0 TD
(\))Tj
ET
276.09 600.166 23.504 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 280.0877 589.7076 Tm
0.0359 Tc
(q)Tj
/F14 1 Tf
7.9701 0 0 7.9701 290.9968 588.0707 Tm
0 Tc
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 303.197 597.1906 Tm
(+)Tj
/F8 1 Tf
1 0 TD
(O)Tj
/F9 1 Tf
0.7905 1.11 TD
( )Tj
/F2 1 Tf
1.0089 -0.428 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 342.8264 602.9948 Tm
(q)Tj
ET
330.45 600.166 20.246 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 330.4409 589.7076 Tm
()Tj
/F2 1 Tf
0.5958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.4824 0 TD
(\))Tj
/F7 1 Tf
0.4984 0.6859 TD
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5716 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(S)Tj
/F9 1 Tf
0.6709 1.11 TD
(\012)Tj
/F8 1 Tf
0.5972 -1.11 TD
(.)Tj
/F2 1 Tf
-22.7726 -2.9751 TD
[(Comparing)-333.3(\(2.5a\))-333.2(and)-333.3(\(2.5b\))-333.2(w)27.8(e)-333.4(deduce)-333.3(that)]TJ
3.2731 -2.3603 TD
(1)Tj
ET
170.37 534.526 17.531 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 170.3851 524.1235 Tm
0.0577 Tc
(S)Tj
/F14 1 Tf
7.9701 0 0 7.9701 183.3514 522.4867 Tm
0 Tc
(q)Tj
1.5884 2.8551 TD
(S)Tj
/F9 1 Tf
10.9091 0 0 10.9091 190.93 541.9696 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 191.4411 518.6521 Tm
(s)Tj
/F1 1 Tf
0.4912 0 TD
(=1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 208.5061 531.6064 Tm
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 213.8471 529.9696 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 218.412 531.6064 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(s)Tj
/F2 1 Tf
0.4688 0 TD
0.2222 Tc
(\)+)Tj
/F8 1 Tf
1.611 0 TD
0 Tc
(O)Tj
/F9 1 Tf
0.7906 1.11 TD
( )Tj
/F8 1 Tf
0.7678 -0.4279 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 276.6954 537.4107 Tm
(q)Tj
ET
261.69 534.526 20.246 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 261.6787 524.1235 Tm
()Tj
/F2 1 Tf
0.5958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
/F9 1 Tf
0.4985 1.7959 TD
(\012)Tj
/F2 1 Tf
0.875 -1.11 TD
(=)Tj
1.7389 0.6765 TD
(1)Tj
ET
305.37 534.526 17.973 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 305.3759 524.1235 Tm
0.0077 Tc
(R)Tj
/F14 1 Tf
7.9701 0 0 7.9701 319.3913 522.3462 Tm
0 Tc
()Tj
/F1 1 Tf
1.9556 2.8727 TD
(2)Tj
/F14 1 Tf
0.5312 0 TD
(R)Tj
/F9 1 Tf
10.9091 0 0 10.9091 332.4294 541.9696 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 326.3622 518.3422 Tm
(r)Tj
/F1 1 Tf
0.509 0 TD
(=)Tj
/F14 1 Tf
0.8264 0 TD
(R)Tj
/F1 1 Tf
0.8065 0 TD
(+1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 356.0717 531.6064 Tm
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 361.4127 529.8292 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 365.3699 531.6064 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(r)Tj
/F2 1 Tf
0.4789 0 TD
0.2223 Tc
(\)+)Tj
/F8 1 Tf
1.6111 0 TD
0 Tc
(O)Tj
/F9 1 Tf
0.7905 1.11 TD
( )Tj
/F8 1 Tf
0.7956 -0.4334 TD
()Tj
/F2 1 Tf
0.4166 0 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 423.3524 537.2095 Tm
()Tj
ET
408.69 534.526 19.531 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 408.7482 524.1235 Tm
()Tj
/F2 1 Tf
0.5958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4166 0 TD
(\))Tj
/F9 1 Tf
0.4985 1.7959 TD
(\012)Tj
/F8 1 Tf
0.5973 -1.11 TD
(.)Tj
/F2 1 Tf
-29.0757 0 TD
(\(2.6\))Tj
2.0091 -3.0796 TD
[(W)83.3(rite)]TJ
/F8 1 Tf
2.9768 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 178.5325 496.2332 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 182.4897 498.0105 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(r)Tj
/F2 1 Tf
0.479 0 TD
0.6048 Tc
(\)=)Tj
/F9 1 Tf
2.3763 0.75 TD
0 Tc
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 229.3948 494.7856 Tm
(d)Tj
/F15 1 Tf
0.5467 0 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
(r)Tj
/F8 1 Tf
10.9091 0 0 10.9091 242.4777 498.0105 Tm
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 247.6812 496.2332 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 251.6384 498.0105 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
[(\))-529.6(for)-529.5(a)-529.6(m)27.8(ultiplicativ)27.9(e)-529.5(function)]TJ
/F8 1 Tf
14.2894 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 422.6471 496.2332 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 426.6053 498.0105 Tm
[(.)-1033.1(Note)-529.5(that)]TJ
/F8 1 Tf
-28.2155 -1.4399 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 124.003 480.526 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 127.9601 482.3024 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 137.6917 486.2615 Tm
(k)Tj
/F2 1 Tf
10.9091 0 0 10.9091 142.8114 482.3024 Tm
0.5347 Tc
[(\)=0)47.2(i)534.7(f)]TJ
/F8 1 Tf
4.2944 0 TD
0 Tc
(p)Tj
/F17 1 Tf
1.0378 0 TD
()Tj
/F8 1 Tf
0.8125 0 TD
()Tj
/F2 1 Tf
0.4166 0 TD
[(.)-907(W)83.3(e)-487.5(also)-487.5(c)27.8(hec)27.8(k)-487.5(easily)-487.5(that)]TJ
/F7 1 Tf
13.3001 0 TD
(|)Tj
/F8 1 Tf
0.2777 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 367.7151 480.526 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 371.6732 482.3024 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 381.4037 486.2615 Tm
(k)Tj
/F2 1 Tf
10.9091 0 0 10.9091 386.5236 482.3024 Tm
(\))Tj
/F7 1 Tf
0.3889 0 TD
0.5347 Tc
(|)Tj
/F2 1 Tf
2.125 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.8281 0 TD
0.3251 Tc
[(+1)325.1(\))]TJ
/F8 1 Tf
1.9917 0 TD
0 Tc
(/)Tj
/F2 1 Tf
0.5 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F7 1 Tf
0.8281 0 TD
()Tj
/F2 1 Tf
1.1028 0 TD
(1\))Tj
-33.0837 -1.2694 TD
[(for)-333.3(primes)]TJ
/F8 1 Tf
4.7611 0 TD
(p)Tj
/F7 1 Tf
0.5031 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
()Tj
/F2 1 Tf
0.4167 0 TD
[(,)-333.3(and)-333.3(note)-333.3(that)]TJ
/F8 1 Tf
6.9444 0 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 265.2095 466.6779 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 272.1963 468.4543 Tm
(=)Tj
/F9 1 Tf
1.0556 0.75 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 295.2269 473.4814 Tm
()Tj
/F14 1 Tf
0 -1.0354 TD
(d)Tj
/F1 1 Tf
0.5468 0 TD
(=1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 312.7213 468.4543 Tm
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 317.9248 466.6779 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 321.882 468.4543 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5206 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/d)Tj
/F2 1 Tf
1.0205 0 TD
0.4445 Tc
[(.T)444.5(h)472.3(u)444.5(s)]TJ
-16.4572 -2.4059 TD
0 Tc
(1)Tj
ET
161.37 437.806 17.973 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 161.3848 427.3445 Tm
0.0077 Tc
(R)Tj
/F14 1 Tf
7.9701 0 0 7.9701 175.4002 425.5672 Tm
0 Tc
()Tj
/F1 1 Tf
1.9556 2.8729 TD
(2)Tj
/F14 1 Tf
0.5312 0 TD
(R)Tj
/F9 1 Tf
10.9091 0 0 10.9091 188.4384 445.1915 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 182.3711 421.5632 Tm
(r)Tj
/F1 1 Tf
0.509 0 TD
(=)Tj
/F14 1 Tf
0.8264 0 TD
(R)Tj
/F1 1 Tf
0.8065 0 TD
(+1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 212.0807 434.8274 Tm
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 217.4217 433.0501 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 221.3788 434.8274 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(r)Tj
/F2 1 Tf
0.4789 0 TD
0.1667 Tc
(\)=)Tj
2.1834 0.6765 TD
0 Tc
(1)Tj
ET
248.37 437.806 17.973 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 248.4065 427.3445 Tm
0.0077 Tc
(R)Tj
/F14 1 Tf
7.9701 0 0 7.9701 262.4219 425.5672 Tm
0 Tc
()Tj
/F9 1 Tf
10.9091 0 0 10.9091 272.3168 445.1915 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 269.3928 421.4745 Tm
(d)Tj
/F15 1 Tf
0.5467 0 TD
()Tj
/F1 1 Tf
0.8264 0 TD
(2)Tj
/F14 1 Tf
0.5313 0 TD
(R)Tj
/F8 1 Tf
10.9091 0 0 10.9091 292.8159 434.8274 Tm
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 298.0194 433.0501 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 301.9766 434.8274 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F9 1 Tf
0.3889 1.1101 TD
( )Tj
/F8 1 Tf
0.7069 -0.4335 TD
(R)Tj
ET
323.85 437.806 8.368 -0.436 re
f
BT
10.9091 0 0 10.9091 325.1955 427.3445 Tm
(d)Tj
/F2 1 Tf
0.9755 0.6859 TD
(+)Tj
/F8 1 Tf
1 0 TD
(O)Tj
/F2 1 Tf
0.7906 0 TD
(\(1\))Tj
/F9 1 Tf
1.2777 1.1101 TD
(\012)Tj
/F2 1 Tf
-12.1369 -4.349 TD
0.1667 Tc
[(=1)-55.5(+)]TJ
/F8 1 Tf
2.6667 0 TD
0 Tc
(O)Tj
/F9 1 Tf
0.7906 1.11 TD
( )Tj
/F2 1 Tf
0.897 -0.4335 TD
(1)Tj
ET
282.33 402.406 9.605 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 282.3333 392.0099 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 287.9811 390.2326 Tm
()Tj
/F9 1 Tf
10.9091 0 0 10.9091 297.876 409.8571 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 294.9529 386.14 Tm
(d>)Tj
/F1 1 Tf
1.3731 0 TD
(2)Tj
/F14 1 Tf
0.5313 0 TD
(R)Tj
/F7 1 Tf
10.9091 0 0 10.9091 319.5716 406.8732 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 327.8047 405.0959 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 331.7629 406.8732 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
(|)Tj
ET
1 i
319.53 401.97 29.384 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 331.4252 392.0099 Tm
(d)Tj
/F2 1 Tf
1.9388 0.6859 TD
(+)Tj
1.6833 0.6765 TD
(1)Tj
ET
364.65 401.97 17.973 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 364.6805 392.0099 Tm
0.0076 Tc
(R)Tj
/F14 1 Tf
7.9701 0 0 7.9701 378.6959 390.2326 Tm
0 Tc
()Tj
/F9 1 Tf
10.9091 0 0 10.9091 388.5908 409.8571 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 385.6667 386.14 Tm
(d)Tj
/F15 1 Tf
0.5468 0 TD
()Tj
/F1 1 Tf
0.8264 0 TD
(2)Tj
/F14 1 Tf
0.5312 0 TD
(R)Tj
/F7 1 Tf
10.9091 0 0 10.9091 409.0899 399.4929 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 417.324 397.7155 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 421.2812 399.4929 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F9 1 Tf
0.2778 1.11 TD
(\012)Tj
/F8 1 Tf
0.5972 -1.11 TD
(.)Tj
/F2 1 Tf
-29.9007 -3.0793 TD
[(W)83.3(e)-333.3(see)-333.4(easily)-333.2(that)-333.3(the)-333.4(error)-333.3(terms)-333.3(ab)-27.7(o)27.8(v)27.8(e)-333.4(are)-333.3(b)-27.7(ounded)-333.3(b)27.8(y)]TJ
/F7 1 Tf
5.6195 -2.7965 TD
()Tj
/F2 1 Tf
2.2489 0.6765 TD
(1)Tj
ET
195.21 338.326 24.252 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 195.2379 325.8251 Tm
(R)Tj
/F12 1 Tf
4.9813 0 0 4.9813 204.8 333.3828 Tm
(1)Tj
ET
204.81 332.263 3.39 -0.373 re
f
BT
4.9813 0 0 4.9813 204.8 326.9988 Tm
(3)Tj
/F8 1 Tf
10.9091 0 0 10.9091 209.8839 325.8251 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 215.5318 324.0478 Tm
()Tj
/F15 1 Tf
1.332 3.1344 TD
()Tj
/F9 1 Tf
10.9091 0 0 10.9091 222.5036 345.7564 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 222.7935 322.0403 Tm
(d)Tj
/F1 1 Tf
0.5466 0 TD
(=1)Tj
/F7 1 Tf
10.9091 0 0 10.9091 241.2742 342.7736 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 249.5083 340.9962 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 253.4655 342.7736 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
ET
241.29 338.326 29.384 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 249.9875 325.8251 Tm
(d)Tj
/F12 1 Tf
4.9813 0 0 4.9813 256.8618 333.3828 Tm
(2)Tj
ET
256.89 332.263 3.39 -0.373 re
f
BT
4.9813 0 0 4.9813 256.8618 326.9988 Tm
(3)Tj
/F7 1 Tf
10.9091 0 0 10.9091 274.8842 335.3933 Tm
()Tj
/F2 1 Tf
1.8087 0.6765 TD
(1)Tj
ET
290.01 338.326 14.647 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 290.0194 325.8251 Tm
(R)Tj
/F12 1 Tf
4.9813 0 0 4.9813 299.5826 333.3828 Tm
(1)Tj
ET
299.61 332.263 3.39 -0.373 re
f
BT
4.9813 0 0 4.9813 299.5826 326.9988 Tm
(3)Tj
/F9 1 Tf
10.9091 0 0 10.9091 307.6802 345.7564 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 309.612 321.597 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
()Tj
/F9 1 Tf
10.9091 0 0 10.9091 323.4371 347.5019 Tm
( )Tj
/F2 1 Tf
0.5973 -1.11 TD
0.2222 Tc
(1+)Tj
/F8 1 Tf
1.7222 0 TD
0 Tc
(O)Tj
/F9 1 Tf
0.7906 1.1099 TD
( )Tj
/F2 1 Tf
0.9962 -0.4334 TD
(1)Tj
ET
365.13 337.89 11.768 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 365.075 325.8251 Tm
(p)Tj
/F12 1 Tf
4.9813 0 0 4.9813 371.7599 333.3828 Tm
(2)Tj
ET
371.73 332.263 3.39 -0.373 re
f
BT
4.9813 0 0 4.9813 371.7599 326.9988 Tm
(3)Tj
/F9 1 Tf
10.9091 0 0 10.9091 378.0384 347.5019 Tm
(\012\012)Tj
/F7 1 Tf
1.4722 -1.11 TD
()Tj
/F2 1 Tf
1.8087 0.6765 TD
(1)Tj
ET
409.29 338.326 14.647 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 409.2344 325.8251 Tm
(R)Tj
/F12 1 Tf
4.9813 0 0 4.9813 418.7975 333.3828 Tm
(1)Tj
ET
418.77 332.263 3.39 -0.373 re
f
BT
4.9813 0 0 4.9813 418.7975 326.9988 Tm
(4)Tj
/F8 1 Tf
10.9091 0 0 10.9091 425.076 335.3933 Tm
(,)Tj
/F2 1 Tf
-28.0753 -3.2851 TD
(since)Tj
/F8 1 Tf
2.45 0 TD
()Tj
/F7 1 Tf
0.6944 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
[(,)-333.4(and)]TJ
/F8 1 Tf
2.5556 0 TD
(R)Tj
/F7 1 Tf
1.0448 0 TD
()Tj
1.2778 0.7203 TD
()Tj
ET
233.13 307.846 6.235 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 233.1597 299.5557 Tm
(x)Tj
/F2 1 Tf
0.5715 0 TD
[(.)-444.4(W)83.3(e)-333.4(conclude)-333.2(that)]TJ
-0.2417 -2.35 TD
(1)Tj
ET
230.49 269.446 17.973 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 230.4976 259.0555 Tm
0.0077 Tc
(R)Tj
/F14 1 Tf
7.9701 0 0 7.9701 244.5131 257.2782 Tm
0 Tc
()Tj
/F1 1 Tf
1.9555 2.8729 TD
(2)Tj
/F14 1 Tf
0.5312 0 TD
(R)Tj
/F9 1 Tf
10.9091 0 0 10.9091 257.5512 276.9026 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 251.4839 253.2742 Tm
(r)Tj
/F1 1 Tf
0.509 0 TD
(=)Tj
/F14 1 Tf
0.8264 0 TD
(R)Tj
/F1 1 Tf
0.8065 0 TD
(+1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 281.1935 266.5385 Tm
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 286.5345 264.7612 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 290.4917 266.5385 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(r)Tj
/F2 1 Tf
0.479 0 TD
0.2778 Tc
[(\)=1)55.6(+)]TJ
/F8 1 Tf
3.4444 0 TD
0 Tc
(O)Tj
/F2 1 Tf
0.7906 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(R)Tj
/F15 1 Tf
7.9701 0 0 7.9701 358.7686 271.0426 Tm
()Tj
/F12 1 Tf
4.9813 0 0 4.9813 366.5504 274.3014 Tm
(1)Tj
ET
366.57 273.223 3.39 -0.373 re
f
BT
4.9813 0 0 4.9813 366.5504 267.9173 Tm
(4)Tj
/F2 1 Tf
10.9091 0 0 10.9091 371.6344 266.5385 Tm
(\))Tj
/F8 1 Tf
0.3889 0 TD
(.)Tj
/F2 1 Tf
-23.5654 -3.0795 TD
[(Com)27.8(bining)-333.3(this)-333.2(with)-333.3(\(2.6\))-333.3(w)27.8(e)-333.4(obtain)-333.3(the)-333.3(prop)-27.8(osition.)]TJ
ET
482.61 239.769 6.775 -0.399 re
f
482.61 239.348 0.399 -5.978 re
f
488.97 239.348 0.399 -5.978 re
f
482.61 233.289 6.775 -0.399 re
f
BT
10.9091 0 0 10.9091 140.7173 207.8137 Tm
[(In)-395.6(Prop)-27.7(osition)-395.5(2.2)-395.5(w)27.8(e)-395.6(compared)-395.4(the)-395.6(distribution)-395.4(of)]TJ
/F7 1 Tf
23.0867 0 TD
(A)Tj
/F2 1 Tf
1.1941 0 TD
[(in)-395.5(t)27.8(w)27.8(o)-395.6(arithmetic)]TJ
-26.2899 -1.2694 TD
[(progressions.)-801.8(W)83.3(e)-452.4(ma)27.8(y)-452.5(also)-452.5(compare)-452.4(the)-452.4(distribution)-452.4(of)]TJ
/F7 1 Tf
25.425 0 TD
(A)Tj
/F2 1 Tf
1.251 0 TD
[(in)-452.4(an)-452.6(arithmetic)]TJ
-26.676 -1.2694 TD
[(progression)-385.4(v)27.7(ersus)-385.4(the)-385.5(distribution)-385.4(in)-385.5(short)-385.5(in)27.8(terv)55.6(als.)-601(Dene)]TJ
27.3415 0.2528 TD
()Tj
-0.1667 -0.2528 TD
(\()Tj
/F8 1 Tf
1.2222 0 TD
(y)Tj
/F2 1 Tf
0.5262 0 TD
0.3647 Tc
(\)=)Tj
2.0628 0.2528 TD
0 Tc
()Tj
-0.1667 -0.2528 TD
(\()Tj
/F8 1 Tf
1.2222 0 TD
0.0359 Tc
[(y,)-130.7(x)]TJ
/F2 1 Tf
1.5422 0 TD
0 Tc
(\))Tj
-33.5838 -1.2694 TD
-0.0278 Tc
(by)Tj
4.277 -2.3292 TD
0 Tc
()Tj
-0.1667 -0.2528 TD
(\()Tj
/F8 1 Tf
1.2223 0 TD
0.0358 Tc
[(y,)-130.9(x)]TJ
/F2 1 Tf
1.5421 0 TD
0 Tc
[(\))-277.8(:)0(=)-2204.6(max)]TJ
/F1 1 Tf
7.9701 0 0 7.9701 215.6144 130.3063 Tm
(\()Tj
/F14 1 Tf
0.4132 0 TD
0.0359 Tc
[(v,)35.9(v)]TJ
/F1 1 Tf
1.4016 0 TD
0 Tc
(+)Tj
/F14 1 Tf
0.8264 0 TD
(y)Tj
/F1 1 Tf
0.561 0 TD
(\))Tj
/F15 1 Tf
0.4133 0 TD
()Tj
/F1 1 Tf
0.8264 0 TD
(\()Tj
/F14 1 Tf
0.4133 0 TD
(x/)Tj
/F1 1 Tf
1.1294 0 TD
(4)Tj
/F14 1 Tf
0.5313 0 TD
(,x)Tj
/F1 1 Tf
0.8932 0 TD
(\))Tj
/F9 1 Tf
10.9091 0 0 10.9091 283.4122 150.6472 Tm
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F7 1 Tf
0.3333 0.05 TD
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(v)Tj
/F2 1 Tf
0.7428 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
(y)Tj
/F2 1 Tf
0.5261 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
0.2222 Tc
(A)Tj
/F2 1 Tf
1.7984 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
(v)Tj
/F2 1 Tf
0.5206 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1 0 TD
(y)Tj
/F7 1 Tf
0.6357 0.6765 TD
(A)Tj
/F2 1 Tf
0.7985 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
ET
385.53 141.046 23.43 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 394.0712 130.6192 Tm
(x)Tj
/F9 1 Tf
1.4692 1.8359 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
0.6667 1.4601 TD
()Tj
/F8 1 Tf
1.3778 -1.41 TD
(y)Tj
/F7 1 Tf
0.6357 0.6765 TD
(A)Tj
/F2 1 Tf
0.7984 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
ET
439.29 141.046 23.43 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 447.9352 130.6192 Tm
(x)Tj
1.4692 0.6859 TD
(.)Tj
/F2 1 Tf
-31.64 0 TD
(\(2.7\))Tj
ET
endstream
endobj
61 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F12 15 0 R
/F14 22 0 R
/F15 23 0 R
/F17 28 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
63 0 obj
<<
/Length 16319
>>
stream
BT
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
(14)Tj
/F5 1 Tf
7.9701 0 0 7.9701 202.119 726.9739 Tm
[(ANDREW)-377.5(GRANVILLE)-377.4(AND)-377.7(K)0(.)-377.7(SOUND)29.4(ARARAJAN)]TJ
10.9091 0 0 10.9091 140.7173 701.071 Tm
[(Pr)22.3(oposition)-423(2.3)]TJ
/F2 1 Tf
7.9009 0 TD
(.)Tj
/F16 1 Tf
1.4261 0 TD
-0.0511 Tc
[(Le)-51.1(t)]TJ
/F8 1 Tf
1.7658 0 TD
0 Tc
(x)Tj
/F16 1 Tf
0.9691 0 TD
[(b)51.1(e)-397.5(lar)51.1(ge)-397.5(and)-397.5(let)]TJ
/F7 1 Tf
7.1485 0 TD
(A)Tj
/F2 1 Tf
0.7985 0 TD
(,)Tj
/F7 1 Tf
0.6853 0 TD
(S)Tj
/F2 1 Tf
0.6805 0 TD
(,)Tj
/F8 1 Tf
0.6853 0 TD
(h)Tj
/F2 1 Tf
0.5762 0 TD
(,)Tj
/F8 1 Tf
0.6853 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 400.4752 699.4351 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 405.0401 701.071 Tm
0.4074 Tc
(,)Tj
/F14 1 Tf
7.9701 0 0 7.9701 421.606 699.4351 Tm
0 Tc
(q)Tj
/F16 1 Tf
10.9091 0 0 10.9091 430.5086 701.071 Tm
(and)Tj
/F2 1 Tf
2.1486 0.2528 TD
()Tj
-0.1667 -0.2528 TD
()Tj
/F16 1 Tf
1.231 0 TD
-0.0511 Tc
[(be)-448.6(a)-51.1(s)]TJ
-31.7862 -1.2694 TD
0 Tc
[(ab)51.1(ove.)-459.9(L)51.1(e)0(t)]TJ
/F8 1 Tf
4.8438 0 TD
(q)Tj
/F7 1 Tf
0.7601 0 TD
()Tj
1.0555 0.7203 TD
()Tj
ET
200.49 695.566 6.235 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 200.539 687.2229 Tm
(x)Tj
/F16 1 Tf
0.9293 0 TD
(with)Tj
/F2 1 Tf
2.1722 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
0.0359 Tc
(q,)Tj
/F7 1 Tf
0.9268 0 TD
0 Tc
(S)Tj
/F2 1 Tf
0.6805 0 TD
0.2778 Tc
(\)=1)Tj
/F16 1 Tf
2.58 0 TD
0 Tc
[(and)-357.7(let)]TJ
/F8 1 Tf
3.3478 0 TD
(y)Tj
/F7 1 Tf
0.8039 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(x/)Tj
/F2 1 Tf
1.0715 0 TD
(4)Tj
/F16 1 Tf
0.8578 0 TD
[(b)51.1(e)-357.8(p)51.1(ositive)-357.7(inte)51.2(gers.)-460(Then)]TJ
/F8 1 Tf
-13.9682 -1.589 TD
(q)Tj
ET
202.29 665.446 20.246 -0.436 re
f
BT
10.9091 0 0 10.9091 202.2765 655.0247 Tm
()Tj
/F2 1 Tf
0.5958 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
0.4984 0.6859 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 232.809 660.8718 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 237.3739 662.5076 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5716 0 TD
0.2222 Tc
(\)+)Tj
1.7777 0.2528 TD
0 Tc
()Tj
-0.1667 -0.2528 TD
(\()Tj
/F8 1 Tf
1.2223 0 TD
[(x,)-166.7(y)]TJ
/F2 1 Tf
1.5421 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F9 1 Tf
1.2778 1.15 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F2 1 Tf
0.924 0.7265 TD
(1)Tj
ET
321.57 665.446 15.953 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 321.6279 655.0247 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 327.2757 653.3888 Tm
(q)Tj
/F8 1 Tf
10.9091 0 0 10.9091 331.8406 655.0247 Tm
(y)Tj
/F9 1 Tf
0.8024 1.636 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 340.9853 649.361 Tm
(s)Tj
/F15 1 Tf
0.4914 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(y)Tj
/F8 1 Tf
10.9091 0 0 10.9091 358.1699 662.5076 Tm
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 363.5109 660.8718 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 368.0757 662.5076 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(s)Tj
/F2 1 Tf
0.4688 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
1 0 TD
(1)Tj
/F9 1 Tf
0.5 1.15 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F8 1 Tf
0.3333 0.05 TD
(.)Tj
/F16 1 Tf
-24.1433 -3.7494 TD
[(Pr)51.2(o)51.1(o)0(f)]TJ
/F2 1 Tf
2.3267 0 TD
[(.)-992.7(The)-350.6(argumen)27.9(t)-350.7(i)0(s)-350.6(similar)-350.5(to)-350.6(the)-350.6(pro)-27.7(of)-350.7(of)-350.6(Prop)-27.7(osition)-350.6(2.2,)-354.9(starting)]TJ
-4.3358 -1.2694 TD
[(with)-443.3(an)]TJ
/F8 1 Tf
3.8866 0 TD
(R)Tj
/F7 1 Tf
1.0626 0 TD
()Tj
/F8 1 Tf
1.0732 0 TD
(y)Tj
/F2 1 Tf
0.9695 0 TD
[(Maier)-443.2(matrix)-443.3(\(again)]TJ
/F8 1 Tf
10.5021 0 TD
(R)Tj
/F2 1 Tf
1.228 0 TD
0.461 Tc
(=[)Tj
/F8 1 Tf
1.5166 0 TD
0 Tc
(x/)Tj
/F2 1 Tf
1.0715 0 TD
(\(4)Tj
/F8 1 Tf
0.8889 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
[(\)]\))-443.3(whose)-443.2(\()]TJ
/F8 1 Tf
4.9477 0 TD
-0.0278 Tc
[(r,)-194.5(s)]TJ
/F2 1 Tf
1.3366 0 TD
0 Tc
(\))Tj
/F1 1 Tf
7.9701 0 0 7.9701 439.0297 611.7151 Tm
(th)Tj
/F2 1 Tf
10.9091 0 0 10.9091 452.3617 607.757 Tm
[(en)27.8(try)-443.2(is)]TJ
-30.5765 -1.2694 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(R)Tj
/F2 1 Tf
0.9892 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
(r)Tj
/F2 1 Tf
0.479 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.7046 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
(s)Tj
/F2 1 Tf
0.4687 0 TD
[(.)-444.5(W)83.4(e)-333.4(omit)-333.3(the)-333.3(details.)]TJ
-3.41 -2.1827 TD
[(W)83.3(e)-444(are)-443.9(nally)-443.9(ready)-443.9(to)-443.9(state)-443.9(our)-443.9(main)-444(general)-443.8(theorems)-443.8(whic)27.8(h)-443.9(will)-444(b)-27.7(e)]TJ
-2.0091 -1.2694 TD
[(pro)27.8(v)27.8(e)0(d)-333.3(i)0(n)-333.3(Section)-333.3(4.)]TJ
/F5 1 Tf
2.0091 -2.1827 TD
[(Theorem)-544.8(2.4)]TJ
/F2 1 Tf
6.4693 0 TD
(.)Tj
/F16 1 Tf
2.1216 0 TD
-0.0511 Tc
[(Le)-51.1(t)]TJ
/F8 1 Tf
1.8725 0 TD
0 Tc
(x)Tj
/F16 1 Tf
1.0757 0 TD
[(b)51.1(e)-504.2(lar)51.2(ge)]TJ
/F2 1 Tf
3.4302 0 TD
(,)Tj
/F16 1 Tf
0.8186 0 TD
[(and)-504.1(in)-504.2(p)51.1(articular)-504.1(supp)51.1(ose)-504.2(that)]TJ
/F7 1 Tf
14.1742 0 TD
0.618 Tc
(S)Tj
/F2 1 Tf
-31.9712 -1.2694 TD
0 Tc
([1)Tj
/F8 1 Tf
0.7778 0 TD
(,)Tj
/F2 1 Tf
0.4444 0 TD
[(log)-180.6(log)]TJ
/F8 1 Tf
2.9167 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(])Tj
/F16 1 Tf
0.2778 0 TD
0.4259 Tc
[(.L)477(e)425.9(t)]TJ
/F2 1 Tf
2.3564 0 TD
0 Tc
(1)Tj
/F8 1 Tf
0.5 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
(100)Tj
/F8 1 Tf
1.7778 0 TD
0.2777 Tc
(> )Tj
/F7 1 Tf
1.8657 0 TD
0 Tc
()Tj
/F2 1 Tf
1.0556 0 TD
[(20)-166.7(log)-180.5(log)-180.6(log)]TJ
/F8 1 Tf
5.5416 0 TD
(x/)Tj
/F2 1 Tf
1.2382 0 TD
[(log)-180.6(log)]TJ
/F8 1 Tf
2.9166 0 TD
(x)Tj
/F16 1 Tf
0.8271 0 TD
[(and)-255.5(supp)51.2(ose)-255.5(that)]TJ
/F2 1 Tf
7.3344 0 TD
(\(log)Tj
/F8 1 Tf
1.8472 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F14 1 Tf
7.9701 0 0 7.9701 486.5375 522.5494 Tm
()Tj
/F7 1 Tf
10.9091 0 0 10.9091 118.7995 504.7433 Tm
()Tj
/F8 1 Tf
1.0555 0 TD
(z)Tj
/F7 1 Tf
0.7868 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(\(log)Tj
/F8 1 Tf
1.8472 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
(3)Tj
/F16 1 Tf
0.8577 0 TD
[(is)-357.8(such)-357.7(that)]TJ
/F9 1 Tf
2.9934 -1.4211 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 216.81 465.4685 Tm
(z)Tj
/F12 1 Tf
4.9813 0 0 4.9813 221.0859 467.737 Tm
(1)Tj
/F11 1 Tf
0.6806 0 TD
()Tj
/F10 1 Tf
1.0834 0 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 234.2545 465.4685 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
(p)Tj
/F15 1 Tf
0.5349 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 258.9798 486.2566 Tm
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F8 1 Tf
1 0 TD
(h)Tj
/F2 1 Tf
0.5762 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
258.93 481.846 39.047 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 275.7589 471.3933 Tm
(p)Tj
/F7 1 Tf
2.4285 0.6859 TD
()Tj
/F8 1 Tf
1.0556 0 TD
( )Tj
/F2 1 Tf
0.6991 0 TD
[(log)-13.8(\(\(1)]TJ
/F7 1 Tf
2.7916 0 TD
()Tj
/F8 1 Tf
1 0 TD
( )Tj
/F2 1 Tf
0.5324 0 TD
(\))Tj
/F15 1 Tf
7.9701 0 0 7.9701 372.807 483.3803 Tm
()Tj
/F1 1 Tf
0.8265 0 TD
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 384.1265 478.8762 Tm
(\))Tj
/F8 1 Tf
0.3889 0 TD
(.)Tj
/F16 1 Tf
-24.7105 -2.9995 TD
[(Then)-357.7(for)-357.7(al)-51.1(l)]TJ
/F2 1 Tf
5.635 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(/ )Tj
/F1 1 Tf
7.9701 0 0 7.9701 196.9893 450.114 Tm
(2)Tj
/F7 1 Tf
10.9091 0 0 10.9091 204.7512 446.1549 Tm
()Tj
/F8 1 Tf
1.0556 0 TD
(u)Tj
/F7 1 Tf
0.8502 0 TD
()Tj
1.0555 0.7203 TD
()Tj
ET
246.09 454.486 5.553 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 246.1479 446.1549 Tm
(z)Tj
/F2 1 Tf
-3.6835 -1.8727 TD
(max)Tj
/F10 1 Tf
4.9813 0 0 4.9813 203.9611 420.1714 Tm
()Tj
/F11 1 Tf
0.5972 0 TD
()Tj
/F10 1 Tf
1.0834 0 TD
-0.0001 Tc
(x/z)Tj
2.2534 0.2946 TD
0 Tc
(u)Tj
/F12 1 Tf
-3.8856 -1.6538 TD
(\()Tj
/F10 1 Tf
0.5416 0 TD
-0.0001 Tc
(,)Tj
/F11 1 Tf
1.0556 0 TD
0 Tc
(S)Tj
/F12 1 Tf
0.9362 0 TD
-0.0002 Tc
(\)=1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 231.2847 425.7255 Tm
0 Tc
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 240.3756 423.9492 Tm
()Tj
/F7 1 Tf
10.9091 0 0 10.9091 247.3624 425.7255 Tm
()Tj
/F2 1 Tf
1.2778 0 TD
(exp)Tj
/F9 1 Tf
1.6944 0.8101 TD
()Tj
/F7 1 Tf
0.4584 -0.8101 TD
()Tj
/F8 1 Tf
0.7778 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
0.2222 Tc
[(\()222.2(1+2)222.2(5)]TJ
/F8 1 Tf
3.1111 0 TD
0 Tc
( )Tj
/F2 1 Tf
0.5324 0 TD
[(\))-166.6(log)-13.9(\(2)]TJ
/F8 1 Tf
2.7361 0 TD
-0.0001 Tc
(u/ )Tj
/F1 1 Tf
7.9701 0 0 7.9701 386.6202 430.2307 Tm
0 Tc
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 391.3524 425.7255 Tm
(\))Tj
/F9 1 Tf
0.389 0.8101 TD
()Tj
/F8 1 Tf
0.6249 -0.8101 TD
(.)Tj
/F2 1 Tf
-23.9888 -3.7855 TD
[(Note)-367.7(that)]TJ
/F9 1 Tf
4.6521 0.75 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 202.9818 381.2035 Tm
(z)Tj
/F12 1 Tf
4.9813 0 0 4.9813 207.2578 383.473 Tm
(1)Tj
/F11 1 Tf
0.6806 0 TD
()Tj
/F10 1 Tf
1.0832 0 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 220.4264 381.2035 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
(p)Tj
/F15 1 Tf
0.5349 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 244.4533 384.4294 Tm
(1)Tj
/F8 1 Tf
0.5 0 TD
(/p)Tj
/F7 1 Tf
1.3382 0 TD
()Tj
/F2 1 Tf
1.1128 0 TD
[(log)-13.9(\(\(1)]TJ
/F7 1 Tf
2.8145 0 TD
()Tj
/F8 1 Tf
1.0229 0 TD
( )Tj
/F2 1 Tf
0.5324 0 TD
(\))Tj
/F15 1 Tf
7.9701 0 0 7.9701 328.5589 388.3876 Tm
()Tj
/F1 1 Tf
0.8265 0 TD
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 339.8785 384.4294 Tm
[(\).)-547.4(There)-367.6(is)-367.7(an)-367.7(analogous)-367.5(result)]TJ
-20.2656 -1.2694 TD
[(for)-333.3(short)-333.2(in)27.8(terv)55.6(als.)]TJ
/F5 1 Tf
2.0091 -2.1826 TD
[(Theorem)-544.8(2.5)]TJ
/F2 1 Tf
6.4693 0 TD
(.)Tj
/F16 1 Tf
2.1216 0 TD
-0.0511 Tc
[(Le)-51.1(t)]TJ
/F8 1 Tf
1.8725 0 TD
0 Tc
(x)Tj
/F16 1 Tf
1.0757 0 TD
[(b)51.1(e)-504.2(lar)51.2(ge)]TJ
/F2 1 Tf
3.4302 0 TD
(,)Tj
/F16 1 Tf
0.8186 0 TD
[(and)-504.1(in)-504.2(p)51.1(articular)-504.1(supp)51.1(ose)-504.2(that)]TJ
/F7 1 Tf
14.1742 0 TD
0.618 Tc
(S)Tj
/F2 1 Tf
-31.9712 -1.2694 TD
0 Tc
([1)Tj
/F8 1 Tf
0.7778 0 TD
(,)Tj
/F2 1 Tf
0.4444 0 TD
[(log)-180.6(log)]TJ
/F8 1 Tf
2.9167 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(])Tj
/F16 1 Tf
0.2778 0 TD
0.4259 Tc
[(.L)477(e)425.9(t)]TJ
/F2 1 Tf
2.3564 0 TD
0 Tc
(1)Tj
/F8 1 Tf
0.5 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
(100)Tj
/F7 1 Tf
1.7778 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
( )Tj
/F7 1 Tf
0.8101 0 TD
()Tj
/F2 1 Tf
1.0556 0 TD
[(20)-166.7(log)-180.5(log)-180.6(log)]TJ
/F8 1 Tf
5.5416 0 TD
(x/)Tj
/F2 1 Tf
1.2382 0 TD
[(log)-180.6(log)]TJ
/F8 1 Tf
2.9166 0 TD
(x)Tj
/F16 1 Tf
0.8271 0 TD
[(and)-255.5(supp)51.2(ose)-255.5(that)]TJ
/F2 1 Tf
7.3344 0 TD
(\(log)Tj
/F8 1 Tf
1.8472 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F14 1 Tf
7.9701 0 0 7.9701 486.5375 336.8807 Tm
()Tj
/F7 1 Tf
10.9091 0 0 10.9091 118.7995 319.0745 Tm
()Tj
/F8 1 Tf
1.0555 0 TD
(z)Tj
/F7 1 Tf
0.7868 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(\(log)Tj
/F8 1 Tf
1.8472 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
(3)Tj
/F16 1 Tf
0.8577 0 TD
[(is)-357.8(such)-357.7(that)]TJ
/F9 1 Tf
2.9934 -1.4211 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 216.81 279.7997 Tm
(z)Tj
/F12 1 Tf
4.9813 0 0 4.9813 221.0859 282.0682 Tm
(1)Tj
/F11 1 Tf
0.6806 0 TD
()Tj
/F10 1 Tf
1.0834 0 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 234.2545 279.7997 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
(p)Tj
/F15 1 Tf
0.5349 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 258.9798 300.5878 Tm
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F8 1 Tf
1 0 TD
(h)Tj
/F2 1 Tf
0.5762 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
258.93 296.206 39.047 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 275.7589 285.7245 Tm
(p)Tj
/F7 1 Tf
2.4285 0.6859 TD
()Tj
/F8 1 Tf
1.0556 0 TD
( )Tj
/F2 1 Tf
0.6991 0 TD
[(log)-13.8(\(\(1)]TJ
/F7 1 Tf
2.7916 0 TD
()Tj
/F8 1 Tf
1 0 TD
( )Tj
/F2 1 Tf
0.5324 0 TD
(\))Tj
/F15 1 Tf
7.9701 0 0 7.9701 372.807 297.7115 Tm
()Tj
/F1 1 Tf
0.8265 0 TD
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 384.1265 293.2075 Tm
(\))Tj
/F8 1 Tf
0.3889 0 TD
(.)Tj
/F16 1 Tf
-24.7105 -2.9995 TD
[(Then)-357.7(for)-357.7(e)51.1(ach)]TJ
/F2 1 Tf
6.4527 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(/ )Tj
/F1 1 Tf
7.9701 0 0 7.9701 205.9099 264.4453 Tm
(2)Tj
/F7 1 Tf
10.9091 0 0 10.9091 213.6727 260.4861 Tm
()Tj
/F8 1 Tf
1.0555 0 TD
(u)Tj
/F7 1 Tf
0.8502 0 TD
()Tj
1.0556 0.7203 TD
()Tj
ET
255.09 268.726 5.553 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 255.0695 260.4861 Tm
(z)Tj
/F16 1 Tf
0.8668 0 TD
[(at)-357.8(le)51.1(ast)-357.7(one)-357.8(of)-357.7(the)-357.7(fol)-51.1(lowing)-357.7(statements)-357.6(is)-357.8(true)]TJ
/F2 1 Tf
20.3278 0 TD
(:)Tj
-32.7415 -2.0389 TD
(\(i\))Tj
/F16 1 Tf
1.5555 0 TD
-0.0767 Tc
[(Fo)-76.7(r)]TJ
/F8 1 Tf
1.9147 0 TD
0 Tc
(q)Tj
/F7 1 Tf
0.8471 0 TD
()Tj
/F8 1 Tf
1.1426 0 TD
(e)Tj
/F1 1 Tf
7.9701 0 0 7.9701 193.7455 242.2027 Tm
(2)Tj
/F14 1 Tf
0.5313 0 TD
(z)Tj
/F16 1 Tf
10.9091 0 0 10.9091 207.1801 238.2436 Tm
[(which)-405.8(is)-405.7(c)51(omp)51.2(ose)51.2(d)-405.9(only)-405.7(of)-405.9(primes)-405.7(in)]TJ
/F2 1 Tf
16.4745 0 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
(z)Tj
/F1 1 Tf
7.9701 0 0 7.9701 395.4849 242.2027 Tm
(1)Tj
/F15 1 Tf
0.5312 0 TD
()Tj
/F14 1 Tf
0.8265 0 TD
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 411.3395 238.2436 Tm
0.1667 Tc
(,z)Tj
/F2 1 Tf
0.9535 0 TD
0.4058 Tc
(]\()Tj
/F16 1 Tf
1.0724 0 TD
0 Tc
[(and)-405.8(so)-405.8(with)]TJ
/F2 1 Tf
-26.3421 -1.2694 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
0.036 Tc
(q,)Tj
/F7 1 Tf
0.9268 0 TD
0 Tc
(S)Tj
/F2 1 Tf
0.6805 0 TD
0.2778 Tc
[(\)=1)277.8(\))]TJ
/F16 1 Tf
2.9689 0 TD
0 Tc
[(and)-357.8(such)-357.7(that)]TJ
/F9 1 Tf
6.2611 0.75 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 280.0548 221.1696 Tm
(p)Tj
/F15 1 Tf
0.5348 0 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 291.2339 224.3955 Tm
(\(1)Tj
/F7 1 Tf
1.1111 0 TD
()Tj
/F8 1 Tf
1 0 TD
(h)Tj
/F2 1 Tf
0.5762 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5032 0 TD
(\)\))Tj
/F8 1 Tf
0.7777 0 TD
(/p)Tj
/F7 1 Tf
1.2809 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
( )Tj
/F1 1 Tf
7.9701 0 0 7.9701 370.0623 228.3547 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 374.7945 224.3955 Tm
(,)Tj
/F16 1 Tf
0.6356 0 TD
[(ther)51.2(e)-357.7(i)0(s)]TJ
/F2 1 Tf
-13.6395 -2.0566 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 242.0244 200.3238 Tm
(q)Tj
/F7 1 Tf
10.9091 0 0 10.9091 249.6189 201.9596 Tm
()Tj
/F2 1 Tf
1.2778 0 TD
(exp\()Tj
/F7 1 Tf
1.9167 0 TD
()Tj
/F8 1 Tf
0.7777 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
0.2222 Tc
[(\()222.2(1+2)222.2(5)]TJ
/F8 1 Tf
3.111 0 TD
0 Tc
( )Tj
/F2 1 Tf
0.5324 0 TD
[(\))-166.7(log)-13.8(\(2)]TJ
/F8 1 Tf
2.7362 0 TD
-0.0001 Tc
(u/ )Tj
/F1 1 Tf
7.9701 0 0 7.9701 386.3014 206.4637 Tm
0 Tc
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 391.0337 201.9596 Tm
(\)\))Tj
/F8 1 Tf
0.7778 0 TD
(.)Tj
/F2 1 Tf
-25.0659 -2.2732 TD
(\(ii\))Tj
/F16 1 Tf
1.8333 0 TD
[(Ther)51.1(e)-357.7(exists)]TJ
/F8 1 Tf
5.6133 0 TD
(y)Tj
/F7 1 Tf
0.8039 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 233.1467 181.1208 Tm
(u)Tj
/F16 1 Tf
10.9091 0 0 10.9091 242.4508 177.1616 Tm
(with)Tj
/F2 1 Tf
2.3389 0.2528 TD
()Tj
-0.1667 -0.2528 TD
(\()Tj
/F8 1 Tf
1.2222 0 TD
(y)Tj
/F2 1 Tf
0.5261 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F2 1 Tf
1.2778 0 TD
(exp\()Tj
/F7 1 Tf
1.9166 0 TD
()Tj
/F8 1 Tf
0.7778 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
0.2222 Tc
[(\()222.2(1+2)222.2(5)]TJ
/F8 1 Tf
3.1111 0 TD
0 Tc
( )Tj
/F2 1 Tf
0.5324 0 TD
[(\))-166.6(log)-13.9(\(2)]TJ
/F8 1 Tf
2.7362 0 TD
(u/ )Tj
/F1 1 Tf
7.9701 0 0 7.9701 429.1756 181.1208 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 433.9079 177.1616 Tm
(\)\))Tj
/F16 1 Tf
0.7778 0 TD
(.)Tj
-27.6536 -2.0389 TD
[(De)51.1(duction)-435.9(of)-435.9(Cor)51.1(ol)-51(lary)]TJ
/F2 1 Tf
10.5985 0 TD
[(1.3.)-699.5(W)83.3(e)-418.4(see)-418.3(readily)-418.4(that)-418.3(there)-418.3(exists)-418.3(\(log)]TJ
/F8 1 Tf
18.7458 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F14 1 Tf
7.9701 0 0 7.9701 471.3156 158.8782 Tm
()Tj
/F7 1 Tf
10.9091 0 0 10.9091 480.9246 154.919 Tm
()Tj
/F8 1 Tf
-33.1948 -1.2694 TD
(z)Tj
/F7 1 Tf
0.7868 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(\(log)Tj
/F8 1 Tf
1.8472 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
[(3)-295.4(satisfying)-295.3(the)-295.3(h)27.8(y)0(p)-27.8(othesis)-295.3(of)-295.3(Theorem)-295.3(2.4.)-431.8(Applying)-295.2(Theorem)-295.3(2.4)]TJ
-5.15 -1.2694 TD
(\(with)Tj
/F8 1 Tf
2.6331 0 TD
(u/ )Tj
/F2 1 Tf
1.9046 0 TD
[(there)-299.7(instead)-299.7(of)]TJ
/F8 1 Tf
7.0465 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
[(\))-299.7(w)27.7(e)-299.7(nd)-299.7(that)-299.7(there)-299.7(exists)]TJ
/F8 1 Tf
11.479 0 TD
()Tj
/F7 1 Tf
0.6944 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(x/z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 412.9744 131.182 Tm
-0.0001 Tc
(u/)Tj
/F7 1 Tf
10.9091 0 0 10.9091 430.1749 127.2229 Tm
0 Tc
()Tj
/F8 1 Tf
1.0556 0 TD
(x/)Tj
/F2 1 Tf
1.0715 0 TD
(\(log)Tj
/F8 1 Tf
1.8472 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F14 1 Tf
7.9701 0 0 7.9701 484.009 131.182 Tm
(u)Tj
ET
endstream
endobj
64 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F10 13 0 R
/F11 14 0 R
/F12 15 0 R
/F14 22 0 R
/F15 23 0 R
/F16 24 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
66 0 obj
<<
/Length 16083
>>
stream
BT
/F5 1 Tf
7.9701 0 0 7.9701 162.3741 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
[(AN)-377.7(UNCER)87.7(T)87.5(AINTY)-377.5(PRINCIPLE)-377.6(F)29.2(O)0(R)-377.8(ARITHMETIC)-377.4(SEQUENCES)]TJ
/F2 1 Tf
10.9091 0 0 10.9091 478.5006 726.9741 Tm
(15)Tj
-32.9726 -2.3745 TD
[(with)-443.4(\()]TJ
/F8 1 Tf
2.7767 0 TD
(,)Tj
/F7 1 Tf
0.8611 0 TD
(S)Tj
/F2 1 Tf
0.6805 0 TD
[(\))-461.2(=)-461.2(1)-443.4(and)-443.3()]TJ
/F14 1 Tf
7.9701 0 0 7.9701 230.4927 699.2936 Tm
()Tj
/F7 1 Tf
10.9091 0 0 10.9091 239.4809 701.071 Tm
()Tj
/F2 1 Tf
1.4612 0 TD
(exp\()Tj
/F7 1 Tf
1.9167 0 TD
()Tj
/F2 1 Tf
0.7777 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
-0.0001 Tc
(u/ )Tj
/F2 1 Tf
1.6048 0 TD
0 Tc
[(\)\(1)-295.5(+)-295.6(2)0(5)]TJ
/F8 1 Tf
3.6468 0 TD
( )Tj
/F2 1 Tf
0.5323 0 TD
[(\))-166.7(log)-13.9(\(2)]TJ
/F8 1 Tf
2.7362 0 TD
(u/ )Tj
/F1 1 Tf
7.9701 0 0 7.9701 399.5118 705.0302 Tm
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 404.2441 701.071 Tm
[(\)\).)-774.5(The)-443.3(corollary)]TJ
-26.1658 -1.2694 TD
[(follo)27.8(ws)-333.4(easily)83.5(.)]TJ
/F16 1 Tf
2.0091 -1.8173 TD
[(De)51.1(duction)-299.2(of)-299.4(Cor)51.2(ol)-51.1(lary)]TJ
/F2 1 Tf
10.1886 0 TD
[(1.4.)-423.2(W)83.4(e)-269.9(ma)27.8(y)-269.8(nd)-269.8(\(log)]TJ
/F8 1 Tf
9.5244 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F14 1 Tf
7.9701 0 0 7.9701 366.2456 671.3564 Tm
()Tj
/F7 1 Tf
10.9091 0 0 10.9091 374.3094 667.3973 Tm
()Tj
/F8 1 Tf
1.0555 0 TD
(z)Tj
/F7 1 Tf
0.7868 0 TD
()Tj
/F2 1 Tf
1.0556 0 TD
(\(log)Tj
/F8 1 Tf
1.8472 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F1 1 Tf
7.9701 0 0 7.9701 436.551 671.3564 Tm
(1)Tj
/F15 1 Tf
0.5312 0 TD
()Tj
/F14 1 Tf
0.8265 0 TD
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 455.3494 667.3973 Tm
(satisfy-)Tj
-30.8504 -1.2694 TD
[(ing)-265.9(the)-265.8(h)27.7(y)0(p)-27.7(othesis)-265.8(of)-265.9(Theorem)-265.8(2.5.)-421.9(The)-265.9(corollary)-265.8(follo)27.8(ws)-265.9(easily)-265.8(b)27.8(y)-265.9(applyication)]TJ
T*
[(of)-333.3(Theorem)-333.3(2.5)-333.3(with)]TJ
/F8 1 Tf
9.2528 0 TD
-0.0001 Tc
(u/ )Tj
/F2 1 Tf
1.9382 0 TD
0 Tc
[(there)-333.3(in)-333.3(place)-333.3(of)]TJ
/F8 1 Tf
7.4195 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
(.)Tj
/F6 1 Tf
-16.4972 -2.9132 TD
[(3.)-575(Oscillations)-383.1(in)-383.3(mean-v)64(alues)-383.3(of)-383.4(m)31.9(ultiplicativ)32.1(e)-383.3(functions)]TJ
/F2 1 Tf
-0.6766 -2.1826 TD
(3a.)Tj
/F16 1 Tf
1.738 0 TD
[(L)51.1(a)0(r)51.1(g)0(e)-362.6(oscil)-51.1(lations.)]TJ
/F2 1 Tf
8.3883 0 TD
[(Throughout)-338.4(this)-338.5(section)-338.6(w)27.8(e)-338.6(shall)-338.5(assume)-338.5(that)]TJ
/F8 1 Tf
20.3174 0 TD
(z)Tj
/F2 1 Tf
0.8476 0 TD
(is)Tj
-33.3004 -1.2694 TD
[(large,)-314.8(and)-310.2(that)]TJ
/F8 1 Tf
6.7715 0 TD
(q)Tj
/F2 1 Tf
0.7925 0 TD
[(is)-310.3(an)-310.2(in)27.8(teger)-310.2(all)-310.2(of)-310.3(whose)-310.1(prime)-310.2(factors)-310.2(are)]TJ
/F7 1 Tf
18.7368 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
[(.)-436.7(Let)]TJ
/F8 1 Tf
2.4831 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 455.215 568.6257 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 459.7798 570.2615 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(n)Tj
/F2 1 Tf
0.6002 0 TD
0.3103 Tc
[(\)b)282.6(e)]TJ
-32.2457 -1.2694 TD
0 Tc
[(a)-377.4(m)27.8(ultiplicativ)27.9(e)-377.3(function)-377.3(with)]TJ
/F8 1 Tf
13.5093 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 271.5148 554.7776 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 276.0797 556.4135 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 285.8102 560.3726 Tm
(k)Tj
/F2 1 Tf
10.9091 0 0 10.9091 290.93 556.4135 Tm
[(\))-351.1(=)-351.1(1)-377.3(for)-377.3(all)]TJ
/F8 1 Tf
5.7536 0 TD
(p)Tj
/F17 1 Tf
0.8542 0 TD
()Tj
/F8 1 Tf
0.6289 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
[(,)-388.3(and)-377.3(0)]TJ
/F7 1 Tf
3.5056 0 TD
()Tj
/F8 1 Tf
1.1289 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 431.0356 554.7776 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 435.6005 556.4135 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(n)Tj
/F2 1 Tf
0.6002 0 TD
(\))Tj
/F7 1 Tf
0.74 0 TD
()Tj
/F2 1 Tf
1.1289 0 TD
[(1)-377.3(for)]TJ
-31.8981 -1.2694 TD
(all)Tj
/F8 1 Tf
1.3889 0 TD
(n)Tj
/F2 1 Tf
0.6003 0 TD
[(.)-444.4(Note)-333.3(that)]TJ
/F8 1 Tf
5.3056 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 203.7191 540.9295 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 208.2839 542.5654 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(n)Tj
/F2 1 Tf
0.6003 0 TD
0.2778 Tc
(\)=)Tj
/F8 1 Tf
1.7222 0 TD
0 Tc
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 243.203 540.9295 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 247.7679 542.5654 Tm
(\(\()Tj
/F8 1 Tf
0.7778 0 TD
[(n,)-166.6(q)]TJ
/F2 1 Tf
1.5269 0 TD
[(\)\))-333.4(is)-333.2(p)-27.8(erio)-27.7(dic)-777.8(\(mo)-27.7(d)]TJ
/F8 1 Tf
9.0361 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
[(\).)-444.4(Dene)]TJ
/F8 1 Tf
-17.1521 -2.7191 TD
(F)Tj
/F14 1 Tf
7.9701 0 0 7.9701 196.6496 511.2668 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 201.2144 512.9026 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(s)Tj
/F2 1 Tf
0.4688 0 TD
0.1666 Tc
(\)=)Tj
/F15 1 Tf
7.9701 0 0 7.9701 230.6799 526.5395 Tm
0 Tc
()Tj
/F9 1 Tf
10.9091 0 0 10.9091 227.0356 523.2668 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 226.935 499.9482 Tm
(n)Tj
/F1 1 Tf
0.6446 0 TD
(=1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 245.9069 520.3437 Tm
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 251.2478 518.7068 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 255.8127 520.3437 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(n)Tj
/F2 1 Tf
0.6002 0 TD
(\))Tj
ET
245.85 515.806 24.938 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 252.8956 505.4197 Tm
(n)Tj
/F14 1 Tf
7.9701 0 0 7.9701 259.4431 508.5719 Tm
(s)Tj
/F2 1 Tf
10.9091 0 0 10.9091 275.0715 512.9026 Tm
(=)Tj
/F8 1 Tf
1.0555 0 TD
()Tj
/F2 1 Tf
0.5113 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(s)Tj
/F2 1 Tf
0.4688 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 314.3402 511.2668 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 318.9051 512.9026 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(s)Tj
/F2 1 Tf
0.4688 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(,)Tj
/F2 1 Tf
-19.5895 -2.3472 TD
(where)Tj
/F8 1 Tf
6.35 -1.6071 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 196.6496 468.1285 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 201.2144 469.7644 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(s)Tj
/F2 1 Tf
0.4688 0 TD
0.1666 Tc
(\)=)Tj
/F9 1 Tf
1.5001 0.95 TD
0 Tc
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 228.5629 455.9691 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
(q)Tj
/F9 1 Tf
10.9091 0 0 10.9091 242.6919 481.874 Tm
( )Tj
/F2 1 Tf
0.5973 -1.11 TD
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F2 1 Tf
1.3135 0.6765 TD
(1)Tj
ET
269.25 472.726 9.903 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 269.1905 462.2814 Tm
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 274.679 465.4336 Tm
(s)Tj
/F9 1 Tf
10.9091 0 0 10.9091 280.2889 481.874 Tm
(\012 )Tj
/F2 1 Tf
1.1944 -1.11 TD
0.2222 Tc
(1+)Tj
/F8 1 Tf
1.8318 0.6821 TD
0 Tc
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 318.6431 475.5686 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 323.2079 477.2055 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
1 i
313.29 472.29 23.879 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 320.2909 462.2814 Tm
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 325.7794 465.4336 Tm
(s)Tj
/F2 1 Tf
10.9091 0 0 10.9091 340.802 469.7644 Tm
(+)Tj
/F8 1 Tf
1.1096 0.6821 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 358.2466 475.5686 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 362.8115 477.2055 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 372.543 481.1646 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 377.2753 477.2055 Tm
(\))Tj
ET
352.89 472.726 28.612 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 360.1435 462.2814 Tm
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 365.6319 465.4336 Tm
(2)Tj
/F14 1 Tf
0.5313 0 TD
(s)Tj
/F2 1 Tf
10.9091 0 0 10.9091 385.1378 469.7644 Tm
(+)Tj
/F8 1 Tf
1 0 TD
0.1667 Tc
(...)Tj
/F9 1 Tf
1.3333 1.11 TD
0 Tc
(\012)Tj
/F8 1 Tf
0.5972 -1.11 TD
(.)Tj
/F2 1 Tf
-27.3448 -3.0596 TD
[(T)83.3(o)-393.4(start)-393.4(with,)]TJ
/F8 1 Tf
6.6204 0 TD
(F)Tj
/F14 1 Tf
7.9701 0 0 7.9701 198.0364 434.7497 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 206.8932 436.3865 Tm
[(is)-393.4(dened)-393.4(in)-393.4(Re\()]TJ
/F8 1 Tf
7.3665 0 TD
(s)Tj
/F2 1 Tf
0.4688 0 TD
(\))Tj
/F8 1 Tf
0.7668 0 TD
(>)Tj
/F2 1 Tf
1.1557 0 TD
[(1,)-408.4(but)-393.4(note)-393.4(that)-393.4(the)-393.5(ab)-27.7(o)27.8(v)27.8(e)-393.5(furnishes)]TJ
-17.8331 -1.2694 TD
[(a)-379.7(meromorphic)-379.6(con)27.8(tin)27.8(uation)-379.7(to)-379.7(Re\()]TJ
/F8 1 Tf
15.6494 0 TD
(s)Tj
/F2 1 Tf
0.4688 0 TD
(\))Tj
/F8 1 Tf
0.7439 0 TD
(>)Tj
/F2 1 Tf
1.1329 0 TD
[(0.)-583.6(Note)-379.6(also)-379.7(that)]TJ
/F8 1 Tf
8.0894 0 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 409.0042 420.9016 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 417.4426 422.5385 Tm
(=)Tj
/F8 1 Tf
1.1329 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 438.3781 420.9016 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 442.943 422.5385 Tm
[(\(1\))-379.7(in)-379.7(the)]TJ
-29.7132 -1.2694 TD
[(notation)-333.2(of)-333.3(Section)-333.3(2.)-444.4(Dene)]TJ
/F8 1 Tf
9.9403 -2.0834 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
0.2778 Tc
[(\):)277.8(=)]TJ
2.3616 0.6765 TD
0 Tc
(1)Tj
ET
269.37 388.966 10.954 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 269.4217 378.4797 Tm
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 274.9738 381.6309 Tm
(u)Tj
/F9 1 Tf
10.9091 0 0 10.9091 285.867 396.3258 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 283.3893 372.8149 Tm
(n)Tj
/F15 1 Tf
0.6446 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F10 1 Tf
4.9813 0 0 4.9813 299.3893 375.0844 Tm
(u)Tj
/F2 1 Tf
10.9091 0 0 10.9091 304.1026 385.9626 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 313.6857 384.3257 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 318.2506 385.9626 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(n)Tj
/F2 1 Tf
0.6003 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 355.1941 384.3257 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 359.7589 385.9626 Tm
(\(1\)\))Tj
/F8 1 Tf
1.6667 0 TD
(,)Tj
/F2 1 Tf
-23.7546 -2.8852 TD
[(and)-333.3(put)-333.3(for)-333.3(all)-333.4(complex)-333.2(n)27.8(u)0(m)27.7(b)-27.7(ers)]TJ
/F8 1 Tf
14.65 0 TD
()Tj
-10.0849 -2.389 TD
(H)Tj
/F14 1 Tf
7.9701 0 0 7.9701 177.6697 326.7895 Tm
(j)Tj
/F2 1 Tf
10.9091 0 0 10.9091 182.0513 328.4264 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
0.1667 Tc
[(\):)166.7(=)]TJ
/F9 1 Tf
1.7778 0.9499 TD
0 Tc
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 213.5014 314.6301 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 229.7345 335.8665 Tm
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F8 1 Tf
1 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 253.863 334.2306 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 258.4279 335.8665 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
229.77 331.366 42.667 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 248.3238 320.9434 Tm
(p)Tj
2.3167 0.6859 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 279.0854 332.9305 Tm
0.0391 Tc
(/)Tj
/F1 1 Tf
1.2145 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F9 1 Tf
10.9091 0 0 10.9091 305.8889 340.536 Tm
( )Tj
/F2 1 Tf
0.7068 -0.4335 TD
[(log)-13.9(\()]TJ
/F8 1 Tf
1.6805 0 TD
0.044 Tc
[(z/)44(p)]TJ
/F2 1 Tf
1.5121 0 TD
0 Tc
(\))Tj
ET
313.65 331.366 39.072 -0.436 re
f
BT
10.9091 0 0 10.9091 322.404 320.9434 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
1.4258 1.796 TD
(\012)Tj
/F14 1 Tf
7.9701 0 0 7.9701 360.3816 337.8172 Tm
(j)Tj
/F2 1 Tf
10.9091 0 0 10.9091 368.4005 328.4264 Tm
[(for)-333.3(eac)27.8(h)]TJ
/F8 1 Tf
3.7805 0 TD
(j)Tj
/F7 1 Tf
0.7469 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(0)Tj
/F8 1 Tf
0.5 0 TD
(,)Tj
/F2 1 Tf
-28.963 -2.607 TD
(and)Tj
/F8 1 Tf
5.1474 -1.4122 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
0.1667 Tc
[(\):)166.7(=)]TJ
/F9 1 Tf
1.7778 0.95 TD
0 Tc
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 213.5014 270.7835 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 232.1176 291.9601 Tm
(1)Tj
ET
229.77 287.566 10.221 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 229.7345 277.0969 Tm
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 235.2229 280.248 Tm
(2)Tj
/F8 1 Tf
10.9091 0 0 10.9091 241.1507 284.5798 Tm
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 246.6401 289.0839 Tm
(2)Tj
/F14 1 Tf
0.5312 0 TD
0.039 Tc
(/)Tj
/F1 1 Tf
1.2144 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F8 1 Tf
10.9091 0 0 10.9091 277.6767 284.5798 Tm
(.)Tj
/F2 1 Tf
-14.5637 -3.0597 TD
(Let)Tj
/F8 1 Tf
1.7917 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
0.2778 Tc
[(\):)277.8(=)]TJ
/F8 1 Tf
2 0 TD
0 Tc
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 188.7034 249.5651 Tm
(0)Tj
/F2 1 Tf
10.9091 0 0 10.9091 193.4356 251.201 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
0.0001 Tc
(\).)Tj
/F5 1 Tf
-5.7048 -2.1826 TD
0 Tc
[(Theorem)-377.6(3.1)]TJ
/F2 1 Tf
6.3022 0 TD
(.)Tj
/F16 1 Tf
1.1667 0 TD
[(With)-357.7(notation)-357.6(as)-357.8(ab)51.1(ove)]TJ
/F2 1 Tf
10.1221 0 TD
(,)Tj
/F16 1 Tf
0.6356 0 TD
(for)Tj
/F2 1 Tf
1.5972 0 TD
(1)Tj
/F7 1 Tf
0.7778 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
()Tj
/F7 1 Tf
0.7613 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 397.9915 231.6862 Tm
(2)Tj
ET
398.01 230.326 4.234 -0.436 re
f
BT
7.9701 0 0 7.9701 397.9915 223.6284 Tm
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 405.2394 227.3903 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
0.044 Tc
(z,)Tj
/F7 1 Tf
-18.0813 -1.9405 TD
0 Tc
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
0.2778 Tc
(|)Tj
/F2 1 Tf
1.6111 0 TD
0 Tc
(exp\()Tj
/F8 1 Tf
1.9166 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1 0 TD
0.046 Tc
(u)Tj
/F2 1 Tf
1.2782 0 TD
0.2222 Tc
(+5)Tj
/F8 1 Tf
1.5 0 TD
0 Tc
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\)\))Tj
/F8 1 Tf
0.7777 0 TD
(.)Tj
/F16 1 Tf
-24.0607 -1.9405 TD
-0.0511 Tc
[(Le)-51.1(t)]TJ
/F1 1 Tf
7.9701 0 0 7.9701 138.8254 189.3469 Tm
0 Tc
(2)Tj
ET
138.81 187.966 4.234 -0.436 re
f
BT
7.9701 0 0 7.9701 138.8254 181.2902 Tm
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 146.0732 185.0521 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F7 1 Tf
0.7868 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
()Tj
/F7 1 Tf
0.7613 0 TD
()Tj
/F8 1 Tf
1.0556 0 TD
()Tj
/F16 1 Tf
0.9637 0 TD
[(and)-357.7(supp)51.2(ose)-357.8(that)]TJ
/F8 1 Tf
7.6411 0 TD
(H)Tj
/F2 1 Tf
0.9124 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(20)Tj
/F8 1 Tf
1 0 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 354.0075 183.4152 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 358.7397 185.0521 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
0.2222 Tc
[(\)+7)222.2(6)]TJ
/F8 1 Tf
2.6111 0 TD
0 Tc
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
0.0001 Tc
[(\))-222.2(+)-222(20,)]TJ
/F16 1 Tf
3.2467 0 TD
0 Tc
[(so)-357.7(that)]TJ
/F8 1 Tf
-22.5649 -2.091 TD
()Tj
/F2 1 Tf
0.8281 0 TD
(:=)Tj
/F9 1 Tf
1.3333 0.9037 TD
()Tj
ET
237.09 172.486 134.556 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 237.1009 162.2406 Tm
(\(5)Tj
/F8 1 Tf
0.8889 0 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 255.8665 160.6037 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 260.5988 162.2406 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
0.2222 Tc
[(\)+1)222.2(9)]TJ
/F8 1 Tf
2.6111 0 TD
0 Tc
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
0.2223 Tc
[(\)+5)222.3(\))]TJ
/F8 1 Tf
2.5 0 TD
0 Tc
(/H)Tj
/F2 1 Tf
1.4125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(1)Tj
/F8 1 Tf
0.5 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
(2)Tj
/F8 1 Tf
0.5 0 TD
(.)Tj
/F16 1 Tf
-26.012 -1.9405 TD
[(Then)-420.7(ther)51.1(e)-420.6(exist)-420.7(p)51.1(oints)]TJ
/F8 1 Tf
10.6183 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 240.8807 139.4342 Tm
()Tj
/F16 1 Tf
10.9091 0 0 10.9091 252.5549 141.071 Tm
[(in)-420.7(the)-420.7(interval)]TJ
/F2 1 Tf
6.7437 0 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\)\(1)Tj
/F7 1 Tf
1.5456 0 TD
()Tj
/F2 1 Tf
1.0456 0 TD
(2)Tj
/F8 1 Tf
0.5 0 TD
()Tj
/F2 1 Tf
0.5504 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
0.1667 Tc
(,H)Tj
/F2 1 Tf
1.3569 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
[(\)\(1)-267.8(+)-267.9(2)]TJ
/F8 1 Tf
3.0912 0 TD
()Tj
/F2 1 Tf
0.5503 0 TD
(\)])Tj
/F16 1 Tf
1.0874 0 TD
(such)Tj
-32.056 -1.2694 TD
(that)Tj
ET
endstream
endobj
67 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F6 9 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F10 13 0 R
/F14 22 0 R
/F15 23 0 R
/F16 24 0 R
/F17 28 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
69 0 obj
<<
/Length 17874
>>
stream
BT
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
(16)Tj
/F5 1 Tf
7.9701 0 0 7.9701 202.119 726.9739 Tm
[(ANDREW)-377.5(GRANVILLE)-377.4(AND)-377.7(K)0(.)-377.7(SOUND)29.4(ARARAJAN)]TJ
/F8 1 Tf
10.9091 0 0 10.9091 170.3741 698.6162 Tm
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 189.5432 696.9794 Tm
(+)Tj
/F2 1 Tf
10.9091 0 0 10.9091 196.6276 698.6162 Tm
(\))Tj
/F7 1 Tf
0.5556 0 TD
()Tj
/F2 1 Tf
2.6327 0.6764 TD
(1)Tj
ET
214.17 701.566 39.897 -0.436 re
f
BT
10.9091 0 0 10.9091 214.1878 691.1323 Tm
(20)Tj
/F8 1 Tf
1 0 TD
0.046 Tc
(H)Tj
/F2 1 Tf
1.396 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
0.6651 0.686 TD
(exp)Tj
/F7 1 Tf
1.5279 0 TD
({)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F2 1 Tf
0.9124 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1 0 TD
0.046 Tc
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 327.7878 696.9794 Tm
0 Tc
(+)Tj
/F7 1 Tf
10.9091 0 0 10.9091 337.2961 698.6162 Tm
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 362.7278 696.9794 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 367.4601 698.6162 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(,)Tj
/F16 1 Tf
-28.1894 -1.9957 TD
(and)Tj
/F8 1 Tf
4.7277 -1.4123 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 189.5432 659.8018 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 196.6276 661.4376 Tm
(\))Tj
/F7 1 Tf
0.5556 0 TD
0.1667 Tc
()Tj
/F2 1 Tf
3.4105 0.6765 TD
0 Tc
(1)Tj
ET
222.69 664.366 39.897 -0.436 re
f
BT
10.9091 0 0 10.9091 222.673 653.9547 Tm
(20)Tj
/F8 1 Tf
1 0 TD
0.046 Tc
(H)Tj
/F2 1 Tf
1.396 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
0.6651 0.6859 TD
(exp)Tj
/F7 1 Tf
1.5278 0 TD
({)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.611 0 TD
()Tj
/F8 1 Tf
1 0 TD
0.0461 Tc
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 336.272 659.8018 Tm
0 Tc
()Tj
/F7 1 Tf
10.9091 0 0 10.9091 345.7813 661.4376 Tm
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 371.213 659.8018 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 375.9452 661.4376 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(.)Tj
/F2 1 Tf
-26.9581 -3.2775 TD
[(In)-419.9(Section)-419.8(3b)-419.9(\(Prop)-27.7(osition)-419.8(3.8\))-419.9(w)27.8(e)-419.9(will)-419.8(sho)27.8(w)-420(that)-419.9(under)-419.8(certain)-419.9(sp)-27.7(ecial)]TJ
-2.0091 -1.2694 TD
[(circumstances)-333.2(one)-333.3(can)-333.3(reduce)-333.2(length)-333.2(of)-333.3(the)-333.4(range)-333.2(for)]TJ
/F8 1 Tf
23.4111 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 380.4384 610.1988 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 391.1602 611.8347 Tm
[(to)-333.3(2.)]TJ
-22.9573 -1.2694 TD
[(W)83.3(e)-333.3(no)27.7(w)-333.3(record)-333.3(some)-333.3(corollaries)-333.2(of)-333.3(Theorem)-333.3(3.1.)]TJ
/F5 1 Tf
0 -2.1827 TD
[(Cor)22.3(ollar)66.7(y)-387.9(3.2)]TJ
/F2 1 Tf
7.2389 0 TD
(.)Tj
/F16 1 Tf
1.2244 0 TD
-0.0511 Tc
[(Le)-51.1(t)]TJ
/F8 1 Tf
1.7349 0 TD
0 Tc
(z)Tj
/F15 1 Tf
7.9701 0 0 7.9701 257.5233 578.4757 Tm
()Tj
/F12 1 Tf
4.9813 0 0 4.9813 267.0007 581.7345 Tm
(1)Tj
ET
265.29 580.663 6.781 -0.373 re
f
BT
4.9813 0 0 4.9813 265.3051 575.3505 Tm
-0.0001 Tc
(10)Tj
/F7 1 Tf
10.9091 0 0 10.9091 276.9843 574.1758 Tm
0 Tc
()Tj
/F8 1 Tf
1.0715 0 TD
( )Tj
/F7 1 Tf
0.8262 0 TD
()Tj
/F2 1 Tf
1.0716 0 TD
(1)Tj
/F8 1 Tf
0.5 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
(100)Tj
/F16 1 Tf
1.8666 0 TD
[(and)-366.6(supp)51.1(ose)-366.6(that)]TJ
/F8 1 Tf
7.6676 0 TD
(q)Tj
/F16 1 Tf
0.8489 0 TD
[(is)-366.6(c)51.1(omp)51.2(ose)51.2(d)]TJ
-28.8527 -1.2694 TD
[(of)-357.7(primes)-357.8(in)]TJ
/F2 1 Tf
5.6861 0 TD
([)Tj
/F8 1 Tf
0.2777 0 TD
(z)Tj
/F1 1 Tf
7.9701 0 0 7.9701 189.4127 564.2869 Tm
(1)Tj
/F15 1 Tf
0.5312 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 205.2672 560.3278 Tm
0.1666 Tc
(,z)Tj
/F2 1 Tf
0.9534 0 TD
0 Tc
(])Tj
/F16 1 Tf
0.6356 0 TD
[(and)-357.7(that)]TJ
/F9 1 Tf
3.3117 -1.4104 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 261.2673 520.7821 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 277.5014 542.0184 Tm
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F8 1 Tf
1 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 301.6299 540.3816 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 306.1948 542.0184 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
277.53 537.526 42.667 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 296.0907 527.0944 Tm
(p)Tj
/F7 1 Tf
2.5944 0.6859 TD
()Tj
/F8 1 Tf
1.0556 0 TD
( )Tj
/F1 1 Tf
7.9701 0 0 7.9701 341.7166 539.0825 Tm
(2)Tj
/F8 1 Tf
10.9091 0 0 10.9091 346.4498 534.5774 Tm
(.)Tj
/F16 1 Tf
-20.8679 -3.0594 TD
[(Then)-357.7(for)]TJ
/F7 1 Tf
4.2038 0.7203 TD
()Tj
ET
173.73 509.446 5.553 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 173.7504 501.2025 Tm
(z)Tj
/F7 1 Tf
0.7868 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(u)Tj
/F7 1 Tf
0.8502 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(/ )Tj
/F1 1 Tf
7.9701 0 0 7.9701 231.3564 505.1617 Tm
(2)Tj
/F16 1 Tf
10.9091 0 0 10.9091 239.992 501.2025 Tm
[(ther)51.1(e)-357.8(exist)-357.7(p)51.1(oints)]TJ
/F8 1 Tf
7.7599 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 330.8911 499.5667 Tm
()Tj
/F7 1 Tf
10.9091 0 0 10.9091 341.0063 501.2025 Tm
()Tj
/F2 1 Tf
0.9444 0 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
[(u,)-166.6(u)]TJ
/F2 1 Tf
1.5893 0 TD
0.2222 Tc
[(\()222.2(1+2)222.2(2)]TJ
/F8 1 Tf
3.1111 0 TD
0 Tc
( )Tj
/F2 1 Tf
0.5324 0 TD
(\)])Tj
/F16 1 Tf
1.0245 0 TD
[(such)-357.6(that)]TJ
/F8 1 Tf
-19.8438 -2.2493 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 225.2922 475.0287 Tm
(+)Tj
/F2 1 Tf
10.9091 0 0 10.9091 232.3766 476.6645 Tm
(\))Tj
/F7 1 Tf
0.5593 0 TD
()Tj
/F2 1 Tf
0.9482 0 TD
(exp)Tj
/F9 1 Tf
1.6944 1.11 TD
( )Tj
/F7 1 Tf
0.8195 -1.11 TD
()Tj
/F8 1 Tf
1 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
0.2222 Tc
[(\()222.2(1+2)222.2(5)]TJ
/F8 1 Tf
3.1111 0 TD
0 Tc
( )Tj
/F2 1 Tf
0.5324 0 TD
[(\))-166.6(log)]TJ
/F9 1 Tf
2.0139 1.11 TD
( )Tj
/F2 1 Tf
0.7068 -0.4335 TD
(2)Tj
/F8 1 Tf
0.5 0 TD
(u)Tj
ET
362.85 479.566 11.699 -0.436 re
f
BT
10.9091 0 0 10.9091 363.4072 469.1816 Tm
( )Tj
/F1 1 Tf
7.9701 0 0 7.9701 369.2155 472.3328 Tm
(2)Tj
/F9 1 Tf
10.9091 0 0 10.9091 375.7231 488.7741 Tm
-0.0001 Tc
(\012\012)Tj
/F8 1 Tf
1.1944 -1.11 TD
0 Tc
(,)Tj
/F16 1 Tf
-24.7458 -1.9401 TD
(and)Tj
/F8 1 Tf
8.0047 -1.4123 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 225.2922 438.4568 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 232.3766 440.0927 Tm
(\))Tj
/F7 1 Tf
0.5593 0 TD
0.1704 Tc
()Tj
/F2 1 Tf
1.8927 0 TD
0 Tc
(exp)Tj
/F9 1 Tf
1.6944 1.11 TD
( )Tj
/F7 1 Tf
0.8195 -1.11 TD
()Tj
/F8 1 Tf
1 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
0.2222 Tc
[(\()222.2(1+2)222.2(5)]TJ
/F8 1 Tf
3.1111 0 TD
0 Tc
( )Tj
/F2 1 Tf
0.5324 0 TD
[(\))-166.6(log)]TJ
/F9 1 Tf
2.0139 1.11 TD
( )Tj
/F2 1 Tf
0.7068 -0.4335 TD
(2)Tj
/F8 1 Tf
0.5 0 TD
(u)Tj
ET
373.17 443.086 11.699 -0.436 re
f
BT
10.9091 0 0 10.9091 373.7106 432.6097 Tm
( )Tj
/F1 1 Tf
7.9701 0 0 7.9701 379.5188 435.7619 Tm
(2)Tj
/F9 1 Tf
10.9091 0 0 10.9091 386.0264 452.2023 Tm
-0.0001 Tc
(\012\012)Tj
/F8 1 Tf
1.1944 -1.11 TD
0 Tc
(.)Tj
/F16 1 Tf
-23.6811 -4.0391 TD
[(Pr)51.2(o)51.1(o)0(f)]TJ
/F2 1 Tf
2.3267 0 TD
[(.)-888.9(Note)-333.2(that)-333.3(for)-333.3(1)]TJ
/F7 1 Tf
8.0584 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
()Tj
/F7 1 Tf
0.7613 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 286.5385 400.3258 Tm
(11)Tj
ET
286.53 398.926 8.468 -0.436 re
f
BT
7.9701 0 0 7.9701 286.5385 392.268 Tm
(20)Tj
/F2 1 Tf
10.9091 0 0 10.9091 298.0204 396.0299 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
-9.7398 -2.4552 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
0.2778 Tc
(\)=)Tj
/F9 1 Tf
1.7222 0.95 TD
0 Tc
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 248.4742 355.449 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 264.7083 376.6864 Tm
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F8 1 Tf
1 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 288.8369 375.0496 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 293.4017 376.6864 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5032 0 TD
(\))Tj
ET
264.69 372.166 42.667 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 283.2976 361.7624 Tm
(p)Tj
2.3167 0.6859 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 314.0593 373.7494 Tm
0.0391 Tc
(/)Tj
/F1 1 Tf
1.2145 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F7 1 Tf
10.9091 0 0 10.9091 343.8924 369.2453 Tm
()Tj
/F8 1 Tf
1.0555 0 TD
( )Tj
/F1 1 Tf
7.9701 0 0 7.9701 361.2155 373.7494 Tm
(2)Tj
/F8 1 Tf
10.9091 0 0 10.9091 365.9477 369.2453 Tm
(e)Tj
/F1 1 Tf
7.9701 0 0 7.9701 371.0276 373.7494 Tm
(\(1)Tj
/F15 1 Tf
0.9444 0 TD
()Tj
/F14 1 Tf
0.8265 0 TD
()Tj
/F1 1 Tf
0.569 0 TD
(\))Tj
/F14 1 Tf
0.4132 0 TD
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 397.5014 369.2453 Tm
(,)Tj
/F2 1 Tf
-25.5477 -2.9201 TD
[(and)-333.3(that)]TJ
/F8 1 Tf
4.8447 -1.8564 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 180.7193 315.5009 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 185.4516 317.1377 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
( )Tj
/F1 1 Tf
7.9701 0 0 7.9701 219.5646 321.6418 Tm
(2)Tj
/F8 1 Tf
10.9091 0 0 10.9091 224.2969 317.1377 Tm
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(,)Tj
/F2 1 Tf
2.4445 0 TD
(and)Tj
/F8 1 Tf
3.6111 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F8 1 Tf
1.0556 0 TD
(e)Tj
/F1 1 Tf
7.9701 0 0 7.9701 354.5544 321.6418 Tm
(2)Tj
/F14 1 Tf
0.5313 0 TD
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 363.3205 317.1377 Tm
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 368.8737 321.6418 Tm
()Tj
/F15 1 Tf
0.569 0 TD
()Tj
/F1 1 Tf
0.8265 0 TD
(1)Tj
/F7 1 Tf
10.9091 0 0 10.9091 387.7579 317.1377 Tm
()Tj
/F8 1 Tf
1.0555 0 TD
( )Tj
/F1 1 Tf
7.9701 0 0 7.9701 405.0809 321.6418 Tm
(2)Tj
/F8 1 Tf
10.9091 0 0 10.9091 409.8132 317.1377 Tm
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(,)Tj
/F2 1 Tf
-28.85 -1.8564 TD
[(where)-358.7(the)-358.7(last)-358.7(inequalit)27.9(y)-358.7(for)]TJ
/F8 1 Tf
12.8047 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
[(\))-358.7(i)0(s)-358.7(easily)-358.6(c)27.7(hec)27.9(k)27.8(e)0(d)-358.7(using)-358.7(our)-358.7(lo)27.8(w)27.7(e)0(r)-358.7(b)-27.7(ound)-358.7(for)]TJ
/F8 1 Tf
-14.3279 -1.2694 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
[(\))-473(and)-473(k)27.8(eeping)-473(in)-473(mind)-473(that)]TJ
/F8 1 Tf
13.0048 0 TD
(z)Tj
/F15 1 Tf
7.9701 0 0 7.9701 285.6956 287.3375 Tm
()Tj
/F12 1 Tf
4.9813 0 0 4.9813 295.1731 290.5952 Tm
(1)Tj
ET
293.49 289.543 6.781 -0.373 re
f
BT
4.9813 0 0 4.9813 293.4774 284.2112 Tm
-0.0001 Tc
(10)Tj
/F7 1 Tf
10.9091 0 0 10.9091 307.5218 283.0376 Tm
0 Tc
()Tj
/F8 1 Tf
1.2884 0 TD
( )Tj
/F7 1 Tf
1.043 0 TD
()Tj
/F2 1 Tf
1.2884 0 TD
(1)Tj
/F8 1 Tf
0.5 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
[(100)-472.9(and)-473(that)]TJ
/F8 1 Tf
6.3635 0 TD
()Tj
/F7 1 Tf
0.9942 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 453.4356 287.3325 Tm
(11)Tj
ET
453.45 286.006 8.468 -0.436 re
f
BT
7.9701 0 0 7.9701 453.4356 279.2757 Tm
(20)Tj
/F2 1 Tf
10.9091 0 0 10.9091 464.9176 283.0376 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.5089 0 TD
(.)Tj
-33.6948 -1.2694 TD
[(F)83.4(rom)-372.6(these)-372.5(estimates)-372.5(it)-372.5(follo)27.8(ws)-372.5(that)-372.6(if)]TJ
/F8 1 Tf
17.2302 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.7321 0 TD
()Tj
/F2 1 Tf
1.1209 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(/ )Tj
/F1 1 Tf
7.9701 0 0 7.9701 363.1691 273.1476 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 371.9661 269.1895 Tm
(then)Tj
/F8 1 Tf
2.317 0 TD
()Tj
/F2 1 Tf
0.923 0 TD
[(\(in)-372.6(Theorem)-372.4(3.1\))]TJ
-26.4469 -1.2694 TD
(is)Tj
/F7 1 Tf
1.0799 0 TD
()Tj
/F2 1 Tf
1.1795 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
( )Tj
/F2 1 Tf
0.5324 0 TD
[(.)-667.7(Therefore)-407.6(from)-407.7(Theorem)-407.7(3.1)-407.7(w)27.8(e)-407.7(conclude)-407.7(that)-407.6(if)]TJ
/F8 1 Tf
22.9406 0 TD
(H)Tj
/F2 1 Tf
0.9124 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.7907 0 TD
()Tj
/F2 1 Tf
1.1795 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(/ )Tj
/F1 1 Tf
7.9701 0 0 7.9701 462.6541 259.2996 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 471.8336 255.3414 Tm
(and)Tj
/F8 1 Tf
-32.3615 -1.2694 TD
()Tj
/F7 1 Tf
0.9257 0 TD
()Tj
/F8 1 Tf
1.0974 0 TD
()Tj
/F7 1 Tf
0.8033 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 162.8005 245.7883 Tm
(11)Tj
ET
162.81 244.486 8.468 -0.436 re
f
BT
7.9701 0 0 7.9701 162.8005 237.7315 Tm
(20)Tj
/F2 1 Tf
10.9091 0 0 10.9091 174.2824 241.4934 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.8676 0 TD
[(then)-358.4(there)-358.4(exist)-358.5(p)-27.8(oin)27.9(ts)]TJ
/F8 1 Tf
10.309 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 318.3631 239.8566 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 329.3589 241.4934 Tm
[(in)-358.4([)]TJ
/F8 1 Tf
1.4696 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\)\(1)Tj
/F7 1 Tf
1.5167 0 TD
()Tj
/F2 1 Tf
1.0168 0 TD
(10)Tj
/F8 1 Tf
1 0 TD
( )Tj
/F2 1 Tf
0.5324 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
0.1667 Tc
(,H)Tj
/F2 1 Tf
1.357 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
[(\)\(1)-238.9(+)-239(1)0(0)]TJ
/F8 1 Tf
3.5335 0 TD
( )Tj
/F2 1 Tf
0.5324 0 TD
(\)])Tj
-33.3059 -1.2694 TD
[(suc)27.8(h)-333.3(that)]TJ
/F8 1 Tf
3.2253 -1.9994 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 173.1537 204.1982 Tm
(+)Tj
/F2 1 Tf
10.9091 0 0 10.9091 180.2391 205.8341 Tm
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F2 1 Tf
2.7437 0.6765 TD
(1)Tj
ET
200.25 208.726 39.897 -0.436 re
f
BT
10.9091 0 0 10.9091 200.2222 198.3512 Tm
(20)Tj
/F8 1 Tf
1 0 TD
0.046 Tc
(H)Tj
/F2 1 Tf
1.396 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
0.6651 0.6859 TD
(exp\()Tj
/F8 1 Tf
1.9167 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1 0 TD
0.046 Tc
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 312.6097 204.1982 Tm
0 Tc
(+)Tj
/F7 1 Tf
10.9091 0 0 10.9091 322.118 205.8341 Tm
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 347.5497 204.1982 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 352.283 205.8341 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\)\))Tj
/F7 1 Tf
1.0555 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(e)Tj
/F15 1 Tf
7.9701 0 0 7.9701 429.5552 210.3382 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
0.0391 Tc
(u)Tj
/F12 1 Tf
4.9813 0 0 4.9813 445.0779 209.3419 Tm
0 Tc
(+)Tj
/F8 1 Tf
10.9091 0 0 10.9091 451.194 205.8341 Tm
(,)Tj
/F2 1 Tf
-30.4695 -2.4846 TD
(and)Tj
/F8 1 Tf
1.8513 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 158.1649 177.0939 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 165.2493 178.7297 Tm
(\))Tj
/F7 1 Tf
0.6667 0 TD
0.2777 Tc
()Tj
/F8 1 Tf
1.8333 0 TD
0 Tc
(e)Tj
/F15 1 Tf
7.9701 0 0 7.9701 197.601 182.689 Tm
()Tj
/F14 1 Tf
0.8265 0 TD
0.039 Tc
(u)Tj
/F11 1 Tf
4.9813 0 0 4.9813 213.1248 181.6926 Tm
0 Tc
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 219.5168 178.7297 Tm
[(.)-413.5(Renaming)]TJ
/F8 1 Tf
5.3342 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\)\(1)Tj
/F7 1 Tf
1.3136 0 TD
()Tj
/F2 1 Tf
0.8137 0 TD
(10)Tj
/F8 1 Tf
1 0 TD
( )Tj
/F2 1 Tf
0.5324 0 TD
0.2778 Tc
(\)=)Tj
/F8 1 Tf
1.7222 0 TD
0 Tc
(u)Tj
/F2 1 Tf
0.8126 0 TD
[(so)-240.2(that)]TJ
/F8 1 Tf
3.2081 0 TD
()Tj
/F7 1 Tf
0.7613 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 426.3303 183.0246 Tm
(1)Tj
ET
420.81 181.726 15.356 -0.436 re
f
BT
7.9701 0 0 7.9701 420.7701 174.9678 Tm
(1)Tj
/F15 1 Tf
0.5312 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 439.1392 178.7297 Tm
[(log)-13.9(\(2)]TJ
/F8 1 Tf
2.1806 0 TD
(u/ )Tj
/F1 1 Tf
7.9701 0 0 7.9701 480.4354 182.689 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 485.1677 178.7297 Tm
(\))Tj
-33.5838 -1.2694 TD
[(w)27.8(e)-333.4(easily)-333.2(obtain)-333.3(the)-333.3(corollary)83.4(.)]TJ
ET
482.61 171.609 6.775 -0.399 re
f
482.61 171.308 0.399 -5.978 re
f
488.97 171.308 0.399 -5.978 re
f
482.61 165.249 6.775 -0.399 re
f
BT
/F5 1 Tf
10.9091 0 0 10.9091 140.7173 141.071 Tm
[(Cor)22.3(ollar)66.7(y)-392.3(3.3)]TJ
/F2 1 Tf
7.2435 0 TD
(.)Tj
/F16 1 Tf
1.25 0 TD
[(Supp)51.2(ose)-370.6(that)]TJ
/F8 1 Tf
5.8777 0 TD
(q)Tj
/F16 1 Tf
0.8529 0 TD
[(is)-370.6(divisible)-370.4(only)-370.5(by)-370.5(primes)-370.4(b)51(etwe)51.2(en)]TJ
/F7 1 Tf
15.3971 0.7203 TD
()Tj
ET
483.81 149.326 5.553 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 483.8566 141.071 Tm
(z)Tj
/F16 1 Tf
-33.4636 -1.2694 TD
(and)Tj
/F8 1 Tf
1.9824 0 TD
(z)Tj
/F16 1 Tf
0.509 0 TD
[(.)-580.4(F)76.6(urther)-397.8(supp)51.2(ose)]TJ
/F8 1 Tf
8.2389 0 TD
(c)Tj
/F16 1 Tf
0.8307 0 TD
[(is)-398(a)-397.9(p)51.1(ositive)-397.8(c)51(onstant)-397.8(such)-397.8(that)-397.9(for)]TJ
/F2 1 Tf
15.7292 0 TD
(1)Tj
/F7 1 Tf
0.8505 0 TD
()Tj
/F8 1 Tf
1.1283 0 TD
()Tj
/F7 1 Tf
0.834 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 460.6994 131.5178 Tm
(2)Tj
ET
460.65 130.126 4.234 -0.436 re
f
BT
7.9701 0 0 7.9701 460.6994 123.461 Tm
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 467.9472 127.2229 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
ET
endstream
endobj
70 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F11 14 0 R
/F12 15 0 R
/F14 22 0 R
/F15 23 0 R
/F16 24 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
72 0 obj
<<
/Length 20956
>>
stream
BT
/F5 1 Tf
7.9701 0 0 7.9701 162.3741 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
[(AN)-377.7(UNCER)87.7(T)87.5(AINTY)-377.5(PRINCIPLE)-377.6(F)29.2(O)0(R)-377.8(ARITHMETIC)-377.4(SEQUENCES)]TJ
/F2 1 Tf
10.9091 0 0 10.9091 478.5006 726.9741 Tm
(17)Tj
/F16 1 Tf
-32.9726 -2.3745 TD
[(ther)51.2(e)-432.6(i)0(s)]TJ
/F8 1 Tf
3.7146 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.8022 0 TD
()Tj
/F8 1 Tf
1.1911 0 TD
(ce)Tj
/F14 1 Tf
7.9701 0 0 7.9701 210.3402 705.0302 Tm
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 214.8722 701.071 Tm
(/)Tj
/F16 1 Tf
0.9835 0 TD
[(.)-684.6(Then)-432.5(ther)51.1(e)-432.6(i)0(s)-432.6(a)-432.6(p)51.1(ositive)-432.5(c)51.1(onstant)]TJ
/F8 1 Tf
16.0706 0 TD
(A)Tj
/F2 1 Tf
1.1826 0 TD
(\()Tj
/F16 1 Tf
0.3889 0 TD
[(dep)51.1(ending)-432.5(only)]TJ
-27.4322 -1.2694 TD
(on)Tj
/F8 1 Tf
1.5223 0 TD
(c)Tj
/F2 1 Tf
0.4327 0 TD
(\))Tj
/F16 1 Tf
0.8378 0 TD
[(such)-448.9(that)-448.9(for)-448.8(al)-51.1(l)]TJ
/F8 1 Tf
7.7118 0 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 238.4747 691.1821 Tm
(A)Tj
/F7 1 Tf
10.9091 0 0 10.9091 250.147 687.2229 Tm
()Tj
/F8 1 Tf
1.2206 0 TD
(u)Tj
/F7 1 Tf
1.0153 0 TD
()Tj
/F8 1 Tf
1.4429 0 TD
(cz)Tj
/F1 1 Tf
7.9701 0 0 7.9701 300.5549 691.1821 Tm
(2)Tj
/F14 1 Tf
0.5312 0 TD
(/)Tj
/F1 1 Tf
0.5312 0 TD
(3)Tj
/F8 1 Tf
10.9091 0 0 10.9091 313.7554 687.2229 Tm
(/)Tj
/F2 1 Tf
0.6667 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.5089 0 TD
(,)Tj
/F16 1 Tf
0.7495 0 TD
[(the)-448.9(interval)]TJ
/F2 1 Tf
5.5106 0 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\(1)Tj
/F7 1 Tf
1.1772 0 TD
()Tj
/F8 1 Tf
1.066 0 TD
(A/)Tj
/F2 1 Tf
1.4167 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(,)Tj
-33.6948 -1.2694 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
[(\(1)-222.2(+)]TJ
/F8 1 Tf
2.1111 0 TD
(A/)Tj
/F2 1 Tf
1.4167 0 TD
(log)Tj
/F8 1 Tf
1.4583 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
(\)])Tj
/F16 1 Tf
1.0244 0 TD
[(c)51.1(ontains)-357.6(p)51.1(oints)]TJ
/F8 1 Tf
6.9 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 278.3761 671.739 Tm
()Tj
/F16 1 Tf
10.9091 0 0 10.9091 289.3639 673.3749 Tm
(satisfying)Tj
/F8 1 Tf
-9.5316 -1.7389 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 204.5519 652.7682 Tm
(+)Tj
/F2 1 Tf
10.9091 0 0 10.9091 211.6364 654.405 Tm
(\))Tj
/F7 1 Tf
0.5556 0 TD
()Tj
/F2 1 Tf
0.9445 0 TD
(exp)Tj
/F7 1 Tf
1.5278 0 TD
({)Tj
/F8 1 Tf
1.2778 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 264.8528 652.7682 Tm
(+)Tj
/F2 1 Tf
10.9091 0 0 10.9091 271.9373 654.405 Tm
(\(log)Tj
/F8 1 Tf
1.8472 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 298.3333 652.7682 Tm
(+)Tj
/F2 1 Tf
10.9091 0 0 10.9091 307.8426 654.405 Tm
[(+)-222.2(log)-180.5(log)]TJ
/F8 1 Tf
3.9166 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 356.815 652.7682 Tm
(+)Tj
/F2 1 Tf
10.9091 0 0 10.9091 366.3233 654.405 Tm
(+)Tj
/F8 1 Tf
1 0 TD
(O)Tj
/F2 1 Tf
0.7906 0 TD
(\(1\)\))Tj
/F7 1 Tf
1.6667 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(,)Tj
/F16 1 Tf
-26.6469 -1.5434 TD
(and)Tj
/F8 1 Tf
6.1035 -1.2694 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 204.5519 622.0832 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 211.6364 623.7201 Tm
(\))Tj
/F7 1 Tf
0.5556 0 TD
0.1667 Tc
()Tj
/F2 1 Tf
1.889 0 TD
0 Tc
(exp)Tj
/F7 1 Tf
1.5278 0 TD
({)Tj
/F8 1 Tf
1.2778 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 275.1552 622.0832 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 282.2396 623.7201 Tm
(\(log)Tj
/F8 1 Tf
1.8473 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 308.6366 622.0832 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 318.145 623.7201 Tm
[(+)-222.2(log)-180.6(log)]TJ
/F8 1 Tf
3.9167 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 367.1173 622.0832 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 376.6267 623.7201 Tm
(+)Tj
/F8 1 Tf
1 0 TD
(O)Tj
/F2 1 Tf
0.7905 0 TD
(\(1\)\))Tj
/F7 1 Tf
1.6667 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(.)Tj
/F2 1 Tf
-25.5822 -2.4695 TD
[(The)-333.3(implied)-333.3(constan)27.9(ts)-333.4(ab)-27.8(o)27.9(v)27.8(e)-333.4(dep)-27.8(end)-333.3(only)-333.3(on)]TJ
/F8 1 Tf
19.9278 0 TD
(c)Tj
/F2 1 Tf
0.4328 0 TD
[(.)-444.4(Note)-333.3(that)]TJ
/F9 1 Tf
-10.4027 -0.789 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 240.7054 569.2742 Tm
()Tj
ET
247.77 569.609 4.276 -0.359 re
f
BT
/F14 1 Tf
7.9701 0 0 7.9701 247.7619 563.5198 Tm
(z)Tj
/F15 1 Tf
0.5365 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(p)Tj
/F15 1 Tf
0.5349 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 275.5676 577.8092 Tm
(1)Tj
/F8 1 Tf
0.5 0 TD
(/p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 291.9651 582.3134 Tm
(1)Tj
/F15 1 Tf
0.5313 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
0.0391 Tc
(/)Tj
/F1 1 Tf
1.2145 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F7 1 Tf
10.9091 0 0 10.9091 332.6187 577.8092 Tm
(\015)Tj
/F8 1 Tf
1.0555 0 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 349.2134 582.3134 Tm
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 353.7445 577.8092 Tm
[(/)-46.1(,)]TJ
/F2 1 Tf
-21.5366 -3.0112 TD
[(b)27.8(y)-360.2(the)-360.2(prime)-360.1(n)27.7(u)0(m)27.8(b)-27.7(er)-360.2(theorem.)-524.9(Th)27.8(us)]TJ
/F8 1 Tf
16.6176 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.7114 0 TD
()Tj
/F8 1 Tf
1.3226 0 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 346.8224 548.9186 Tm
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 351.3544 544.9594 Tm
(/)Tj
/F2 1 Tf
0.9835 0 TD
[(,)-367(and)-360.1(the)-360.2(criterion)]TJ
/F8 1 Tf
8.3975 0 TD
(H)Tj
/F2 1 Tf
0.9124 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.7114 0 TD
()Tj
/F8 1 Tf
-33.1948 -1.2694 TD
(ce)Tj
/F14 1 Tf
7.9701 0 0 7.9701 128.5997 535.0705 Tm
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 133.1317 531.1113 Tm
(/)Tj
/F2 1 Tf
1.3546 0 TD
[(in)-371.1(Corollary)-371.1(3.3)-371.1(ma)27.8(y)-371.1(b)-27.8(e)-371.1(l)0(o)-27.8(osely)-371.1(in)27.8(terpreted)-371(as)-371.2(sa)27.8(ying)-371.1(that,)-380.5(t)27.8(ypically,)]TJ
-2.6684 -1.2694 TD
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F8 1 Tf
1 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 142.928 515.6274 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 147.4929 517.2633 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F8 1 Tf
1.2777 0 TD
(c)Tj
/F2 1 Tf
0.4328 0 TD
(.)Tj
-3.8903 -1.2694 TD
(If)Tj
/F8 1 Tf
1 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 201.2503 507.3744 Tm
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 205.7823 503.4152 Tm
(/)Tj
/F2 1 Tf
1.3169 0 TD
[(then)-333.2(our)-333.3(b)-27.8(ounds)-333.3(tak)27.8(e)-333.4(the)-333.3(shap)-27.7(e)]TJ
-1.339 -1.7389 TD
(exp)Tj
/F7 1 Tf
1.5279 0 TD
({)Tj
/F8 1 Tf
1.2777 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
[(\(log)-13.9(\()]TJ
/F8 1 Tf
2.0694 0 TD
-0.0001 Tc
(u/)Tj
/F2 1 Tf
1.6486 0 TD
0 Tc
[(\))-222.3(+)-222.2(log)-180.6(log)]TJ
/F8 1 Tf
4.5278 0 TD
(u)Tj
/F7 1 Tf
0.7947 0 TD
()Tj
/F2 1 Tf
1 0 TD
0.2222 Tc
(1+)Tj
/F8 1 Tf
1.7222 0 TD
0 Tc
(o)Tj
/F2 1 Tf
0.4847 0 TD
(\(1\)\))Tj
/F7 1 Tf
1.6667 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(.)Tj
/F16 1 Tf
-23.7343 -2.6521 TD
[(Pr)51.2(o)51.1(o)0(f)-370.7(o)0(f)-370.6(Cor)51.1(ol)-51(lary)]TJ
/F2 1 Tf
8.3453 0 TD
[(3.3.)-973.1(In)-347.3(this)-347.3(situation)]TJ
/F8 1 Tf
9.9429 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.69 0 TD
()Tj
/F9 1 Tf
1.0789 0.75 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 387.6543 458.0413 Tm
()Tj
ET
394.77 458.369 4.276 -0.359 re
f
BT
/F14 1 Tf
7.9701 0 0 7.9701 394.7108 452.287 Tm
(z)Tj
/F15 1 Tf
0.5365 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(p)Tj
/F15 1 Tf
0.5349 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F8 1 Tf
10.9091 0 0 10.9091 423.0147 455.5128 Tm
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 428.5031 459.471 Tm
(2)Tj
/F14 1 Tf
0.5312 0 TD
0.0391 Tc
(/)Tj
/F1 1 Tf
1.2144 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F8 1 Tf
10.9091 0 0 10.9091 459.5397 455.5128 Tm
(/p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 470.4827 459.471 Tm
(2)Tj
/F7 1 Tf
10.9091 0 0 10.9091 478.5006 455.5128 Tm
()Tj
/F8 1 Tf
-32.9726 -1.4275 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 123.8795 443.8994 Tm
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 128.4104 439.9403 Tm
(/)Tj
/F7 1 Tf
0.5 0.7203 TD
()Tj
ET
143.01 448.246 5.553 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 142.9559 439.9403 Tm
(z)Tj
/F2 1 Tf
0.7312 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
(e)Tj
/F1 1 Tf
7.9701 0 0 7.9701 166.922 443.8994 Tm
(2)Tj
/F14 1 Tf
0.5312 0 TD
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 175.6882 439.9403 Tm
(/z)Tj
/F7 1 Tf
1.2868 0 TD
()Tj
/F8 1 Tf
1.2778 0 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 208.7452 443.8994 Tm
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 213.2762 439.9403 Tm
(/)Tj
/F1 1 Tf
7.9701 0 0 7.9701 224.006 443.8994 Tm
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 228.7382 439.9403 Tm
[(.)-444.4(F)83.4(urther,)-333.3(b)27.8(y)-333.4(the)-333.3(prime)-333.2(n)27.7(u)0(m)27.8(b)-27.7(er)-333.4(theorem,)]TJ
/F8 1 Tf
-8.6921 -2.3982 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 142.9828 412.1415 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 147.7151 413.7784 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F9 1 Tf
1.8479 0.9499 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 176.0199 405.2423 Tm
()Tj
ET
183.09 405.569 4.276 -0.359 re
f
BT
/F14 1 Tf
7.9701 0 0 7.9701 183.0775 399.4879 Tm
(z)Tj
/F15 1 Tf
0.5364 0 TD
()Tj
/F14 1 Tf
0.8265 0 TD
(p)Tj
/F15 1 Tf
0.5348 0 TD
()Tj
/F14 1 Tf
0.8265 0 TD
(z)Tj
/F8 1 Tf
10.9091 0 0 10.9091 212.0777 421.1586 Tm
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 217.5671 425.1168 Tm
0.039 Tc
(/)Tj
/F1 1 Tf
1.2144 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
ET
212.13 416.686 32.291 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 225.4794 406.2953 Tm
(p)Tj
/F9 1 Tf
1.8412 1.7959 TD
( )Tj
/F2 1 Tf
0.7068 -0.4334 TD
[(log)-13.9(\()]TJ
/F8 1 Tf
1.6805 0 TD
0.044 Tc
[(z/)44(p)]TJ
/F2 1 Tf
1.5121 0 TD
0 Tc
(\))Tj
ET
253.29 416.686 39.072 -0.436 re
f
BT
10.9091 0 0 10.9091 262.0802 406.2953 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
1.4258 1.7959 TD
(\012)Tj
/F1 1 Tf
7.9701 0 0 7.9701 300.0578 423.1691 Tm
(2)Tj
/F7 1 Tf
10.9091 0 0 10.9091 307.8207 413.7784 Tm
()Tj
/F9 1 Tf
1.2778 1.3611 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 332.6695 425.4715 Tm
(z)Tj
/F15 1 Tf
-0.6084 -1.9864 TD
()Tj
ET
1 i
334.89 409.65 4.276 0.359 re
f
BT
/F14 1 Tf
7.9701 0 0 7.9701 334.8772 403.8854 Tm
(z)Tj
/F8 1 Tf
10.9091 0 0 10.9091 342.665 421.1586 Tm
(t)Tj
/F14 1 Tf
7.9701 0 0 7.9701 346.6042 425.1168 Tm
0.0391 Tc
(/)Tj
/F1 1 Tf
1.2145 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
ET
342.69 416.25 30.742 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 345.2334 406.2953 Tm
(t)Tj
/F2 1 Tf
0.5278 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(t)Tj
/F9 1 Tf
0.706 1.7959 TD
( )Tj
/F2 1 Tf
0.7068 -0.4334 TD
[(log)-13.9(\()]TJ
/F8 1 Tf
1.6805 0 TD
0.044 Tc
[(z/)44(t)]TJ
/F2 1 Tf
1.3701 0 TD
0 Tc
(\))Tj
ET
382.29 416.686 37.523 -0.436 re
f
BT
10.9091 0 0 10.9091 390.3442 406.2953 Tm
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F9 1 Tf
1.3548 1.7959 TD
(\012)Tj
/F1 1 Tf
7.9701 0 0 7.9701 427.5467 423.1691 Tm
(2)Tj
/F8 1 Tf
10.9091 0 0 10.9091 432.279 413.7784 Tm
(dt)Tj
/F7 1 Tf
1.1594 0 TD
()Tj
/F8 1 Tf
1.4056 0.6765 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 465.339 425.1168 Tm
()Tj
ET
460.05 416.686 10.007 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 460.0618 406.2953 Tm
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 465.336 409.4466 Tm
(3)Tj
/F8 1 Tf
10.9091 0 0 10.9091 471.2638 413.7784 Tm
(.)Tj
/F2 1 Tf
-32.3092 -2.8911 TD
[(The)-375.3(corollary)-375.2(no)27.7(w)-375.3(follo)27.8(ws)-375.3(from)-375.3(Theorem)-375.2(3.1,)-385.8(and)-375.3(b)27.8(y)-375.3(the)-375.3(fact)-375.2(that)]TJ
/F8 1 Tf
29.7532 0 TD
(u)Tj
/F2 1 Tf
0.9202 0 TD
(=)Tj
/F8 1 Tf
1.1255 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
-33.5838 -1.2694 TD
[(so)-333.3(that)]TJ
/F8 1 Tf
3.3944 0 TD
()Tj
/F2 1 Tf
0.7613 0 TD
[(=)-277.8(log)]TJ
/F8 1 Tf
2.5139 0 TD
(u)Tj
/F2 1 Tf
0.7947 0 TD
[(+)-222.2(log)-180.6(log)]TJ
/F8 1 Tf
3.9166 0 TD
(u)Tj
/F2 1 Tf
0.7947 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
(O)Tj
/F2 1 Tf
0.7906 0 TD
(\(1\).)Tj
ET
482.61 375.129 6.775 -0.399 re
f
482.61 374.708 0.399 -5.978 re
f
488.97 374.708 0.399 -5.978 re
f
482.61 368.769 6.775 -0.398 re
f
BT
/F5 1 Tf
10.9091 0 0 10.9091 140.7173 344.5808 Tm
[(Cor)22.3(ollar)66.7(y)-492.8(3.4)]TJ
/F2 1 Tf
7.3439 0 TD
(.)Tj
/F16 1 Tf
1.8241 0 TD
[(Ke)51.1(ep)-458.6(the)-458.6(notation)-458.4(as)-458.6(in)-458.5(The)51.1(or)51.1(em)]TJ
/F2 1 Tf
15.4164 0 TD
(3.1,)Tj
/F16 1 Tf
2.0394 0 TD
[(and)-458.5(supp)51.2(ose)]TJ
/F8 1 Tf
-28.633 -1.2694 TD
(q)Tj
/F16 1 Tf
0.9806 0 TD
[(is)-498.2(divisible)-498.2(only)-498.2(by)-498.4(the)-498.3(primes)-498.2(b)51.1(etwe)51.1(en)]TJ
/F8 1 Tf
17.9653 0 TD
0.0439 Tc
(z/)Tj
/F2 1 Tf
1.0089 0 TD
0 Tc
(2)Tj
/F16 1 Tf
0.9984 0 TD
(and)Tj
/F8 1 Tf
2.0827 0 TD
(z)Tj
/F16 1 Tf
0.509 0 TD
[(.)-881.5(F)76.6(urther)-498.2(supp)51.1(ose)-498.2(that)]TJ
/F8 1 Tf
-23.545 -1.2694 TD
(c)Tj
/F16 1 Tf
0.9369 0 TD
[(is)-504.2(a)-504.1(p)51.1(ositive)-504.1(c)51(onstant)-504(such)-504.1(that)-504.1(for)]TJ
/F2 1 Tf
16.473 0 TD
(1)Tj
/F7 1 Tf
1.0429 0 TD
()Tj
/F8 1 Tf
1.3207 0 TD
()Tj
/F7 1 Tf
1.0265 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 361.3121 321.1805 Tm
(2)Tj
ET
361.29 319.846 4.234 -0.436 re
f
BT
7.9701 0 0 7.9701 361.3121 313.1228 Tm
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 368.5599 316.8847 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(,)Tj
/F8 1 Tf
0.8185 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.9318 0 TD
()Tj
/F8 1 Tf
1.3208 0 TD
-0.0001 Tc
(ce)Tj
/F14 1 Tf
7.9701 0 0 7.9701 452.798 320.8438 Tm
0 Tc
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 457.33 316.8847 Tm
(/)Tj
/F2 1 Tf
0.6666 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F16 1 Tf
0.509 0 TD
(.)Tj
-33.6659 -1.2694 TD
[(Then)-459.3(ther)51.1(e)-459.4(i)0(s)-459.4(a)-459.3(p)51.1(ositive)-459.3(c)51.1(onstant)]TJ
/F8 1 Tf
15.24 0 TD
(A)Tj
/F2 1 Tf
1.2094 0 TD
(\()Tj
/F16 1 Tf
0.3889 0 TD
[(dep)51.2(ending)-459.4(only)-459.3(on)]TJ
/F8 1 Tf
8.5592 0 TD
(c)Tj
/F2 1 Tf
0.4327 0 TD
(\))Tj
/F16 1 Tf
0.8483 0 TD
[(such)-459.3(that)-459.3(for)-459.3(al)-51.1(l)]TJ
/F8 1 Tf
-26.6785 -1.2694 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 123.8795 293.1477 Tm
(A)Tj
/F7 1 Tf
10.9091 0 0 10.9091 135.0715 289.1885 Tm
()Tj
/F8 1 Tf
1.1767 0 TD
(u)Tj
/F7 1 Tf
0.9713 0 TD
()Tj
/F8 1 Tf
1.3989 0 TD
(cz)Tj
/F1 1 Tf
7.9701 0 0 7.9701 184.0399 293.1477 Tm
(2)Tj
/F14 1 Tf
0.5312 0 TD
(/)Tj
/F1 1 Tf
0.5312 0 TD
(3)Tj
/F8 1 Tf
10.9091 0 0 10.9091 197.2403 289.1885 Tm
(/)Tj
/F2 1 Tf
0.6667 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F16 1 Tf
0.9336 0 TD
[(the)-424.6(interval)]TJ
/F2 1 Tf
5.462 0 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\(1)Tj
/F7 1 Tf
1.1595 0 TD
()Tj
/F8 1 Tf
1.0485 0 TD
(A/)Tj
/F2 1 Tf
1.4166 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\))Tj
/F8 1 Tf
0.389 0 TD
0.1666 Tc
(,u)Tj
/F2 1 Tf
1.0169 0 TD
0 Tc
[(\(1)-270.6(+)]TJ
/F8 1 Tf
2.2079 0 TD
(A/)Tj
/F2 1 Tf
1.4167 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\)])Tj
/F16 1 Tf
1.0913 0 TD
[(c)51.1(ontains)]TJ
-30.3693 -1.2694 TD
[(p)51.1(oints)]TJ
/F8 1 Tf
2.9389 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 157.1049 273.7046 Tm
()Tj
/F16 1 Tf
10.9091 0 0 10.9091 168.0927 275.3405 Tm
(satisfying)Tj
/F8 1 Tf
0.1189 -2.1079 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 188.5589 250.7088 Tm
(+)Tj
/F2 1 Tf
10.9091 0 0 10.9091 195.6443 252.3457 Tm
(\))Tj
/F7 1 Tf
0.5555 0 TD
()Tj
/F2 1 Tf
2.517 0.6765 TD
(1)Tj
ET
213.21 255.286 37.371 -0.436 re
f
BT
10.9091 0 0 10.9091 213.2035 244.8627 Tm
[(log)-180.6(log)]TJ
/F8 1 Tf
2.9167 0 TD
(z)Tj
/F2 1 Tf
0.7853 0.6859 TD
(exp)Tj
/F7 1 Tf
1.5278 0 TD
({)Tj
/F8 1 Tf
1.2778 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 290.4398 250.7088 Tm
(+)Tj
/F2 1 Tf
10.9091 0 0 10.9091 297.5243 252.3457 Tm
(\(log)Tj
/F8 1 Tf
1.8472 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 323.9213 250.7088 Tm
(+)Tj
/F2 1 Tf
10.9091 0 0 10.9091 333.4296 252.3457 Tm
[(+)-222.2(log)-180.6(log)]TJ
/F8 1 Tf
3.9167 0 TD
(z)Tj
/F2 1 Tf
0.7312 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
(O)Tj
/F2 1 Tf
0.7905 0 TD
(\(1\)\))Tj
/F7 1 Tf
1.6667 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(,)Tj
/F16 1 Tf
-28.2795 -1.9401 TD
(and)Tj
/F8 1 Tf
4.6374 -1.4123 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 188.5589 214.1379 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 195.6443 215.7738 Tm
(\))Tj
/F7 1 Tf
0.5555 0 TD
0.1667 Tc
()Tj
/F2 1 Tf
3.2947 0.6765 TD
0 Tc
(1)Tj
ET
221.73 218.686 37.371 -0.436 re
f
BT
10.9091 0 0 10.9091 221.6887 208.2909 Tm
[(log)-180.6(log)]TJ
/F8 1 Tf
2.9167 0 TD
(z)Tj
/F2 1 Tf
0.7852 0.6859 TD
(exp)Tj
/F7 1 Tf
1.5279 0 TD
({)Tj
/F8 1 Tf
1.2777 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 298.925 214.1379 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 306.0095 215.7738 Tm
(\(log)Tj
/F8 1 Tf
1.8472 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 332.4055 214.1379 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 341.9148 215.7738 Tm
[(+)-222.2(log)-180.6(log)]TJ
/F8 1 Tf
3.9166 0 TD
(z)Tj
/F2 1 Tf
0.7313 0 TD
(+)Tj
/F8 1 Tf
0.9999 0 TD
(O)Tj
/F2 1 Tf
0.7906 0 TD
(\(1\)\))Tj
/F7 1 Tf
1.6667 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(.)Tj
/F2 1 Tf
-27.0482 -3.049 TD
[(As)-356.9(in)-356.9(Corollary)-356.8(3.3,)-362.9(the)-356.8(implied)-356.9(constan)27.8(ts)-356.9(ab)-27.7(o)27.8(v)27.8(e)-356.9(dep)-27.8(end)-356.8(only)-356.9(on)]TJ
/F8 1 Tf
28.8155 0 TD
(c)Tj
/F2 1 Tf
0.4327 0 TD
[(.)-515.3(Also)]TJ
-31.2574 -1.2694 TD
[(note)-333.3(that)]TJ
/F8 1 Tf
4.3889 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
[(\))-333.3(i)0(n)-333.3(this)-333.3(case)-333.2(is)-333.3(alw)27.8(a)27.8(y)0(s)]TJ
/F7 1 Tf
10.1611 0 TD
()Tj
/F9 1 Tf
1.0556 0.7501 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 320.0289 165.4396 Tm
0.0423 Tc
(z/)Tj
/F1 1 Tf
1.0678 0 TD
0 Tc
(2)Tj
/F15 1 Tf
0.5312 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(p)Tj
/F15 1 Tf
0.5349 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 356.801 168.6644 Tm
(1)Tj
/F8 1 Tf
0.5 0 TD
(/p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 373.1985 172.6236 Tm
(1)Tj
/F15 1 Tf
0.5313 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
0.0391 Tc
(/)Tj
/F1 1 Tf
1.2145 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F7 1 Tf
10.9091 0 0 10.9091 413.8521 168.6644 Tm
(\015)Tj
/F8 1 Tf
1.0555 0 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 430.4468 172.6236 Tm
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 434.9778 168.6644 Tm
(/)Tj
/F2 1 Tf
0.6667 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(.)Tj
/F16 1 Tf
-29.6079 -2.3531 TD
[(Pr)51.2(o)51.1(o)0(f)-308.7(o)0(f)-308.7(Cor)51.2(ol)-51.1(lary)]TJ
/F2 1 Tf
8.1594 0 TD
[(3.4.)-853.3(In)-279.9(this)-280(case)]TJ
/F8 1 Tf
7.5655 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F8 1 Tf
1.2777 0 TD
(e)Tj
/F1 1 Tf
7.9701 0 0 7.9701 355.1681 146.953 Tm
(2)Tj
/F14 1 Tf
0.5312 0 TD
()Tj
/F9 1 Tf
10.9091 0 0 10.9091 365.7524 151.1761 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 377.2673 139.7689 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
0.0423 Tc
(z/)Tj
/F1 1 Tf
1.0677 0 TD
0 Tc
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 403.1771 142.9948 Tm
(1)Tj
/F8 1 Tf
0.5 0 TD
(/p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 419.5746 146.953 Tm
(2)Tj
/F7 1 Tf
10.9091 0 0 10.9091 427.3375 142.9948 Tm
()Tj
/F8 1 Tf
1.2777 0 TD
(e)Tj
/F1 1 Tf
7.9701 0 0 7.9701 446.3562 146.953 Tm
(2)Tj
/F14 1 Tf
0.5312 0 TD
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 455.1223 142.9948 Tm
(/z)Tj
/F2 1 Tf
1.1756 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F7 1 Tf
-33.4636 -1.4458 TD
()Tj
/F8 1 Tf
1.2778 0 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 137.8182 131.182 Tm
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 142.3502 127.2229 Tm
(/)Tj
/F2 1 Tf
0.6667 0 TD
(log)Tj
/F1 1 Tf
7.9701 0 0 7.9701 163.7141 131.6433 Tm
(3)Tj
/F8 1 Tf
10.9091 0 0 10.9091 170.2645 127.2229 Tm
(z)Tj
/F2 1 Tf
0.509 0 TD
[(.)-407.3(F)83.3(urther)-222.1(note)-222.2(that)]TJ
/F8 1 Tf
8.3712 0 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 276.2092 125.5861 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 280.9415 127.2229 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F2 1 Tf
1.0556 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 318.5684 131.182 Tm
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 323.0994 127.2229 Tm
(/)Tj
/F2 1 Tf
0.6667 0 TD
(log)Tj
/F1 1 Tf
7.9701 0 0 7.9701 344.4633 131.6433 Tm
(2)Tj
/F8 1 Tf
10.9091 0 0 10.9091 351.0137 127.2229 Tm
(z)Tj
/F2 1 Tf
0.509 0 TD
(\))Tj
/F9 1 Tf
0.5555 0.75 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 374.143 123.9971 Tm
0.0422 Tc
(z/)Tj
/F1 1 Tf
1.0676 0 TD
0 Tc
(2)Tj
/F15 1 Tf
0.5312 0 TD
()Tj
/F14 1 Tf
0.8265 0 TD
(p)Tj
/F15 1 Tf
0.5347 0 TD
()Tj
/F14 1 Tf
0.8265 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 410.9141 127.2229 Tm
(1)Tj
/F8 1 Tf
0.5 0 TD
(/p)Tj
/F7 1 Tf
1.2809 0 TD
()Tj
/F8 1 Tf
1.2778 0 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 449.3609 131.182 Tm
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 453.8929 127.2229 Tm
(/)Tj
/F2 1 Tf
0.6667 0 TD
(log)Tj
/F1 1 Tf
7.9701 0 0 7.9701 475.2568 131.6433 Tm
(3)Tj
/F8 1 Tf
10.9091 0 0 10.9091 481.8073 127.2229 Tm
(z)Tj
/F2 1 Tf
0.509 0 TD
(.)Tj
ET
endstream
endobj
73 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F14 22 0 R
/F15 23 0 R
/F16 24 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
75 0 obj
<<
/Length 19391
>>
stream
BT
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
(18)Tj
/F5 1 Tf
7.9701 0 0 7.9701 202.119 726.9739 Tm
[(ANDREW)-377.5(GRANVILLE)-377.4(AND)-377.7(K)0(.)-377.7(SOUND)29.4(ARARAJAN)]TJ
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 701.071 Tm
[(T)83.3(aking)]TJ
/F8 1 Tf
3.2698 0 TD
(u)Tj
/F2 1 Tf
0.8502 0 TD
(=)Tj
/F8 1 Tf
1.0555 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
[(\))-269.7(s)0(o)-269.7(that)]TJ
/F8 1 Tf
3.9258 0 TD
()Tj
/F2 1 Tf
0.7613 0 TD
[(=)-277.7(log)]TJ
/F8 1 Tf
2.5139 0 TD
(u)Tj
/F2 1 Tf
0.6674 0 TD
[(+)-95(log)-180.6(log)]TJ
/F8 1 Tf
3.7895 0 TD
(z)Tj
/F2 1 Tf
0.604 0 TD
(+)Tj
/F8 1 Tf
0.8728 0 TD
(O)Tj
/F2 1 Tf
0.7905 0 TD
[(\(1\))-269.8(and)-269.7(th)27.8(us)]TJ
/F8 1 Tf
5.5647 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F8 1 Tf
1.2778 0 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 453.1128 705.0302 Tm
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 457.6448 701.071 Tm
(/)Tj
/F2 1 Tf
0.6667 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.5089 0 TD
(,)Tj
-33.6948 -1.2694 TD
[(w)27.8(e)-333.4(easily)-333.2(deduce)-333.3(Corollary)-333.3(3.4)-333.3(from)-333.3(Theorem)-333.3(3.1.)]TJ
ET
482.61 693.969 6.775 -0.398 re
f
482.61 693.548 0.399 -5.978 re
f
488.97 693.548 0.399 -5.978 re
f
482.61 687.609 6.775 -0.398 re
f
BT
/F5 1 Tf
10.9091 0 0 10.9091 140.7173 663.4122 Tm
[(Cor)22.3(ollar)66.7(y)-468.6(3.5)]TJ
/F2 1 Tf
7.3196 0 TD
(.)Tj
/F16 1 Tf
1.6858 0 TD
[(Ke)51.2(ep)-437.4(the)-437.4(notation)-437.2(of)-437.3(The)51.1(or)51.1(em)]TJ
/F2 1 Tf
13.8806 0 TD
(3.1,)Tj
/F16 1 Tf
2.0129 0 TD
[(and)-437.3(supp)51.1(ose)]TJ
/F2 1 Tf
5.7558 0 TD
(\()Tj
/F16 1 Tf
0.3889 0 TD
(as)Tj
-33.0526 -1.2694 TD
[(in)-353.2(Cor)51.1(ol)-51(lary)]TJ
/F2 1 Tf
5.6642 0 TD
(3.3\))Tj
/F16 1 Tf
2.0199 0 TD
(that)Tj
/F8 1 Tf
2.0399 0 TD
(q)Tj
/F16 1 Tf
0.8355 0 TD
[(is)-353.2(divisible)-353.1(only)-353.2(by)-353.2(primes)-353.1(b)51(etwe)51.2(en)]TJ
/F7 1 Tf
15.2932 0.7203 TD
()Tj
ET
409.89 657.886 5.553 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 409.9208 649.5641 Tm
(z)Tj
/F16 1 Tf
0.8622 0 TD
(and)Tj
/F8 1 Tf
1.9377 0 TD
(z)Tj
/F16 1 Tf
0.8623 0 TD
[(and)-353.1(that)]TJ
-30.3483 -1.2694 TD
(for)Tj
/F2 1 Tf
1.684 0 TD
(1)Tj
/F7 1 Tf
0.9351 0 TD
()Tj
/F8 1 Tf
1.2129 0 TD
()Tj
/F7 1 Tf
0.9185 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 185.05 640.011 Tm
(2)Tj
ET
185.01 638.686 4.234 -0.436 re
f
BT
7.9701 0 0 7.9701 185.05 631.9542 Tm
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 192.2979 635.7161 Tm
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(,)Tj
/F8 1 Tf
0.7441 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.8239 0 TD
()Tj
/F8 1 Tf
1.2128 0 TD
(ce)Tj
/F14 1 Tf
7.9701 0 0 7.9701 273.3689 639.6752 Tm
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 277.8999 635.7161 Tm
(/)Tj
/F16 1 Tf
0.9836 0 TD
0.7205 Tc
[(.L)771.6(e)720.5(t)]TJ
/F8 1 Tf
2.84 0 TD
0 Tc
(y)Tj
/F2 1 Tf
0.9613 0 TD
(=)Tj
/F8 1 Tf
1.2128 0 TD
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 348.8817 639.6752 Tm
(u)Tj
/F16 1 Tf
10.9091 0 0 10.9091 359.1332 635.7161 Tm
(with)Tj
/F2 1 Tf
2.259 0 TD
(1)Tj
/F7 1 Tf
0.9351 0 TD
()Tj
/F8 1 Tf
1.2129 0 TD
(u)Tj
/F7 1 Tf
1.0075 0 TD
()Tj
/F8 1 Tf
1.4351 0 TD
(cz)Tj
/F1 1 Tf
7.9701 0 0 7.9701 444.1285 639.6752 Tm
(2)Tj
/F14 1 Tf
0.5312 0 TD
(/)Tj
/F1 1 Tf
0.5314 0 TD
(3)Tj
/F8 1 Tf
10.9091 0 0 10.9091 457.33 635.7161 Tm
(/)Tj
/F2 1 Tf
0.6667 0 TD
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F16 1 Tf
0.509 0 TD
(.)Tj
-33.6659 -1.2694 TD
[(Ther)51.2(e)-456.8(i)0(s)-456.8(a)-456.7(p)51.1(ositive)-456.7(c)51(onstant)]TJ
/F8 1 Tf
12.9021 0 TD
(B)Tj
/F2 1 Tf
1.2655 0 TD
(\()Tj
/F16 1 Tf
0.3889 0 TD
[(dep)51.1(ending)-456.6(only)-456.8(on)]TJ
/F8 1 Tf
8.5514 0 TD
(c)Tj
/F2 1 Tf
0.4327 0 TD
(\))Tj
/F16 1 Tf
0.8457 0 TD
[(such)-456.7(that)-456.8(the)-456.7(interval)]TJ
/F2 1 Tf
-24.3862 -1.2694 TD
([1)Tj
/F8 1 Tf
0.7778 0 TD
0.1666 Tc
(,z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 137.6857 611.9791 Tm
0 Tc
(u)Tj
/F1 1 Tf
0.6153 0 TD
(\(1+)Tj
/F14 1 Tf
1.7708 0 TD
0.0493 Tc
(B/)Tj
/F1 1 Tf
1.5591 0 TD
0 Tc
[(log)-14.7(\()]TJ
/F14 1 Tf
1.7856 0 TD
(u)Tj
/F1 1 Tf
0.6151 0 TD
(+1\)\)+)Tj
/F14 1 Tf
3.0105 0 TD
(B)Tj
/F2 1 Tf
10.9091 0 0 10.9091 219.5347 608.02 Tm
(])Tj
/F16 1 Tf
0.6356 0 TD
[(c)51.1(ontains)-357.7(numb)51.2(ers)]TJ
/F8 1 Tf
7.9349 0 TD
(v)Tj
/F15 1 Tf
7.9701 0 0 7.9701 318.3193 606.3841 Tm
()Tj
/F16 1 Tf
10.9091 0 0 10.9091 329.3071 608.02 Tm
(satisfying)Tj
/F2 1 Tf
-18.75 -1.627 TD
(1)Tj
ET
124.65 585.886 5.739 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 124.6187 575.4072 Tm
(y)Tj
/F9 1 Tf
2.3193 1.636 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 133.3728 569.7435 Tm
(v)Tj
/F12 1 Tf
4.9813 0 0 4.9813 137.4964 568.7472 Tm
(+)Tj
/F15 1 Tf
7.9701 0 0 7.9701 143.1143 569.7435 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
(n)Tj
/F15 1 Tf
0.6446 0 TD
()Tj
/F14 1 Tf
0.8265 0 TD
(v)Tj
/F12 1 Tf
4.9813 0 0 4.9813 165.5482 568.7472 Tm
(+)Tj
/F1 1 Tf
7.9701 0 0 7.9701 171.1661 569.7435 Tm
(+)Tj
/F14 1 Tf
0.8265 0 TD
(y)Tj
/F2 1 Tf
10.9091 0 0 10.9091 182.2247 582.8912 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 191.8077 581.2543 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 196.3726 582.8912 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(n)Tj
/F2 1 Tf
0.6002 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 233.3161 581.2543 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 237.881 582.8912 Tm
(\(1\)\))Tj
/F7 1 Tf
1.9445 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(exp)Tj
/F7 1 Tf
1.5279 0 TD
({)Tj
/F8 1 Tf
1.2777 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
[(\(log)-13.9(\()]TJ
/F8 1 Tf
2.0694 0 TD
(u)Tj
/F2 1 Tf
0.7947 0 TD
[(+)-222.2(1)0(\))-222.1(+)-222.2(log)-180.6(log)-13.9(\()]TJ
/F8 1 Tf
6.25 0 TD
(u)Tj
/F2 1 Tf
0.7947 0 TD
0.2222 Tc
[(+2)222.2(\)+)]TJ
/F8 1 Tf
3.1111 0 TD
0 Tc
(O)Tj
/F2 1 Tf
0.7905 0 TD
(\(1\)\))Tj
/F7 1 Tf
1.6667 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(,)Tj
/F16 1 Tf
-33.271 -2.8996 TD
(and)Tj
/F2 1 Tf
0.1226 -1.377 TD
(1)Tj
ET
119.97 531.766 5.739 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 119.995 521.3738 Tm
(y)Tj
/F9 1 Tf
2.3447 1.6359 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 128.7482 515.7091 Tm
(v)Tj
/F11 1 Tf
4.9813 0 0 4.9813 132.8717 514.7128 Tm
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 138.7666 515.7091 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
(n)Tj
/F15 1 Tf
0.6447 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(v)Tj
/F11 1 Tf
4.9813 0 0 4.9813 161.2015 514.7128 Tm
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 167.0964 515.7091 Tm
(+)Tj
/F14 1 Tf
0.8264 0 TD
(y)Tj
/F2 1 Tf
10.9091 0 0 10.9091 178.1539 528.8568 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 187.738 527.2199 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 192.3029 528.8568 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(n)Tj
/F2 1 Tf
0.6003 0 TD
(\))Tj
/F7 1 Tf
0.5963 0 TD
()Tj
/F8 1 Tf
0.9852 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 228.9245 527.2199 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 233.4894 528.8568 Tm
(\(1\)\))Tj
/F7 1 Tf
1.9444 0 TD
0.2778 Tc
()Tj
/F2 1 Tf
2 0 TD
0 Tc
(exp)Tj
/F7 1 Tf
1.5279 0 TD
({)Tj
/F8 1 Tf
1.2777 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
[(\(log)-13.9(\()]TJ
/F8 1 Tf
2.0694 0 TD
(u)Tj
/F2 1 Tf
0.7799 0 TD
[(+)-207.5(1)0(\))-207.4(+)-207.5(log)-180.5(log)-13.9(\()]TJ
/F8 1 Tf
6.2058 0 TD
(u)Tj
/F2 1 Tf
0.78 0 TD
0.2074 Tc
[(+2)207.4(\)+)]TJ
/F8 1 Tf
3.0668 0 TD
0 Tc
(O)Tj
/F2 1 Tf
0.7905 0 TD
(\(1\)\))Tj
/F7 1 Tf
1.6667 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(.)Tj
/F16 1 Tf
-31.6857 -3.8129 TD
[(Pr)51.2(o)51.1(o)0(f)]TJ
/F2 1 Tf
2.3267 0 TD
[(.)-854.5(App)-27.7(ealing)-281.6(to)-281.6(Corollary)-281.6(3.3)-281.7(w)27.8(e)-281.7(see)-281.7(that)-281.7(there)-281.6(is)-281.7(some)]TJ
/F8 1 Tf
23.974 0 TD
(w)Tj
/F2 1 Tf
1.0206 0 TD
(=)Tj
/F8 1 Tf
1.0555 0 TD
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 455.8356 491.2209 Tm
(u)Tj
/F12 1 Tf
4.9813 0 0 4.9813 460.7383 490.2247 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 468.1973 487.2618 Tm
(with)Tj
/F8 1 Tf
-32.0281 -1.2694 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 125.0441 471.7778 Tm
(1)Tj
/F7 1 Tf
10.9091 0 0 10.9091 133.1547 473.4137 Tm
()Tj
/F2 1 Tf
0.9763 0 TD
([)Tj
/F8 1 Tf
0.2777 0 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 151.9143 477.3729 Tm
(A)Tj
/F2 1 Tf
10.9091 0 0 10.9091 161.3181 473.4137 Tm
(+)Tj
/F8 1 Tf
1.0128 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
[(\(1)-234.9(+)]TJ
/F8 1 Tf
2.1366 0 TD
0.0277 Tc
(D/)Tj
/F2 1 Tf
1.5223 0 TD
0 Tc
[(log)-13.9(\()]TJ
/F8 1 Tf
1.6806 0 TD
(u)Tj
/F2 1 Tf
0.8074 0 TD
[(+)-234.9(1\)\))]TJ
/F8 1 Tf
2.2905 0 TD
0.1666 Tc
(,e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 280.5838 477.3729 Tm
0 Tc
(A)Tj
/F2 1 Tf
10.9091 0 0 10.9091 289.9875 473.4137 Tm
(+)Tj
/F8 1 Tf
1.0128 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
[(\(1)-234.9(+)-235(\()]TJ
/F8 1 Tf
2.5255 0 TD
(D)Tj
/F2 1 Tf
1.0906 0 TD
0.235 Tc
(+3)Tj
/F8 1 Tf
1.5128 0 TD
0 Tc
(A)Tj
/F2 1 Tf
0.75 0 TD
(\))Tj
/F8 1 Tf
0.389 0 TD
(/)Tj
/F2 1 Tf
0.6667 0 TD
[(log)-13.8(\()]TJ
/F8 1 Tf
1.6805 0 TD
(u)Tj
/F2 1 Tf
0.8074 0 TD
[(+)-234.9(1\)\)])-352.4(\(here)]TJ
/F8 1 Tf
5.4982 0 TD
(A)Tj
/F2 1 Tf
1.1024 0 TD
(is)Tj
-33.3004 -1.2694 TD
[(as)-333.3(in)-333.3(Corollary)-333.3(3.3)-333.3(and)]TJ
/F8 1 Tf
10.3722 0 TD
(D)Tj
/F2 1 Tf
1.189 0 TD
[(is)-333.3(a)-333.3(suitably)-333.2(large)-333.4(p)-27.7(ositiv)27.8(e)-333.4(constan)27.9(t\))-333.4(suc)27.8(h)-333.3(that)]TJ
/F9 1 Tf
-7.5254 -1.1826 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 161.6986 423.1542 Tm
(n)Tj
/F15 1 Tf
0.6446 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(w)Tj
/F2 1 Tf
10.9091 0 0 10.9091 179.7141 436.3019 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 189.2971 434.665 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 193.862 436.3019 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(n)Tj
/F2 1 Tf
0.6002 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 230.8055 434.665 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 235.3704 436.3019 Tm
(\(1\)\))Tj
/F7 1 Tf
1.9445 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(w)Tj
/F2 1 Tf
0.9095 0 TD
(exp\()Tj
/F7 1 Tf
1.9167 0 TD
()Tj
/F8 1 Tf
0.7778 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 313.6588 434.665 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 318.3911 436.3019 Tm
(\(log)Tj
/F8 1 Tf
1.8472 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 344.787 434.665 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 351.9442 436.3019 Tm
[(+)-222.2(log)-180.6(log)]TJ
/F8 1 Tf
3.9166 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 400.9166 434.665 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 408.0728 436.3019 Tm
(+)Tj
/F8 1 Tf
1 0 TD
(C)Tj
/F1 1 Tf
7.9701 0 0 7.9701 426.7786 434.665 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 431.5109 436.3019 Tm
(\)\))Tj
/F9 1 Tf
0.7778 1.11 TD
(\012)Tj
/F2 1 Tf
-29.443 -3.9654 TD
[(for)-333.3(some)-333.3(absolute)-333.2(constan)27.8(t)]TJ
/F8 1 Tf
12.0195 0 TD
(C)Tj
/F1 1 Tf
7.9701 0 0 7.9701 257.7176 403.5158 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 262.4498 405.1517 Tm
(.)Tj
-11.1588 -1.2694 TD
[(W)83.3(e)-397.9(no)27.8(w)-398(divide)-397.8(the)-397.9(in)27.8(terv)55.6(al)-397.9([1)]TJ
/F8 1 Tf
13.2147 0 TD
0.1667 Tc
(,w)Tj
/F2 1 Tf
1.1873 0 TD
0 Tc
[(])-397.9(i)0(n)27.9(t)0(o)-398(subin)27.8(terv)55.6(als)-397.9(of)-397.9(the)-397.9(form)-397.9(\()]TJ
/F8 1 Tf
14.1542 0 TD
(w)Tj
/F7 1 Tf
1.0081 0 TD
()Tj
/F8 1 Tf
1.0431 0 TD
0.0337 Tc
(ky,)Tj
-32.6166 -1.2694 TD
0 Tc
(w)Tj
/F7 1 Tf
0.9579 0 TD
()Tj
/F2 1 Tf
0.9928 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(k)Tj
/F7 1 Tf
0.7672 0 TD
()Tj
/F2 1 Tf
0.9928 0 TD
(1\))Tj
/F8 1 Tf
0.8889 0 TD
(y)Tj
/F2 1 Tf
0.5262 0 TD
[(])-329.7(for)-329.7(in)27.8(tegers)-329.7(1)]TJ
/F7 1 Tf
6.6114 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(k)Tj
/F7 1 Tf
0.8299 0 TD
()Tj
/F2 1 Tf
1.0556 0 TD
([)Tj
/F8 1 Tf
0.2777 0 TD
0.0269 Tc
[(w/)26.9(y)]TJ
/F2 1 Tf
1.769 0 TD
0 Tc
[(],)-330.5(together)-329.7(with)-329.7(one)-329.7(last)-329.7(in)27.9(terv)55.6(al)-329.7([1)]TJ
/F8 1 Tf
15.2125 0 TD
0.1667 Tc
(,y)Tj
/F1 1 Tf
7.9701 0 0 7.9701 481.6469 375.8197 Tm
0 Tc
(0)Tj
/F2 1 Tf
10.9091 0 0 10.9091 486.3791 377.4555 Tm
(])Tj
-33.6948 -1.2694 TD
(where)Tj
/F8 1 Tf
2.8954 0 TD
(y)Tj
/F1 1 Tf
7.9701 0 0 7.9701 155.734 361.9716 Tm
(0)Tj
/F2 1 Tf
10.9091 0 0 10.9091 163.5646 363.6075 Tm
(=)Tj
/F8 1 Tf
1.0618 0 TD
(w)Tj
/F7 1 Tf
0.9675 0 TD
()Tj
/F2 1 Tf
1.0026 0 TD
([)Tj
/F8 1 Tf
0.2777 0 TD
0.0269 Tc
[(w/)26.9(y)]TJ
/F2 1 Tf
1.769 0 TD
0 Tc
(])Tj
/F8 1 Tf
0.2777 0 TD
(y)Tj
/F2 1 Tf
0.8102 0 TD
(=)Tj
/F8 1 Tf
1.0617 0 TD
(y)Tj
/F7 1 Tf
0.5262 0 TD
({)Tj
/F8 1 Tf
0.5 0 TD
0.0269 Tc
[(w/)26.9(y)]TJ
/F7 1 Tf
1.769 0 TD
0 Tc
(})Tj
/F2 1 Tf
0.5 0 TD
[(.)-455.6(Put)]TJ
/F8 1 Tf
2.6955 0 TD
(y)Tj
/F1 1 Tf
7.9701 0 0 7.9701 313.1198 361.9716 Tm
(0)Tj
/F2 1 Tf
10.9091 0 0 10.9091 320.9505 363.6075 Tm
(=)Tj
/F8 1 Tf
1.0618 0 TD
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 338.0872 367.5667 Tm
(u)Tj
/F12 1 Tf
4.9813 0 0 4.9813 342.9898 366.5704 Tm
(0)Tj
/F2 1 Tf
10.9091 0 0 10.9091 347.3763 363.6075 Tm
[(.)-455.6(Then,)-338(using)-337(the)-337(rst)-337.1(part)-337(of)]TJ
-20.9529 -1.2694 TD
[(Theorem)-310.7(3.1)-310.8(to)-310.7(b)-27.8(ound)]TJ
/F7 1 Tf
10.0515 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 250.6521 348.1235 Tm
(0)Tj
/F2 1 Tf
10.9091 0 0 10.9091 255.3843 349.7594 Tm
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F2 1 Tf
0.5886 0 TD
[(\(taking)-310.7(there)]TJ
/F8 1 Tf
5.9855 0 TD
()Tj
/F2 1 Tf
0.7613 0 TD
[(=)-277.7(log)-13.9(\()]TJ
/F8 1 Tf
2.7362 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 375.742 348.1235 Tm
(0)Tj
/F2 1 Tf
10.9091 0 0 10.9091 382.407 349.7594 Tm
[(+)-177.1(1)0(\))-177.1(+)-177.1(log)-180.6(log)-13.9(\()]TJ
/F8 1 Tf
6.1147 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 455.3584 348.1235 Tm
(0)Tj
/F2 1 Tf
10.9091 0 0 10.9091 462.0224 349.7594 Tm
[(+)-177.2(2\)\),)]TJ
-31.4621 -1.2694 TD
[(w)27.8(e)-333.4(get)-333.3(that)]TJ
/F9 1 Tf
0.9132 -1.2009 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
0.6852 1.0001 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 134.2167 297.1178 Tm
(n)Tj
/F15 1 Tf
0.6446 0 TD
()Tj
/F14 1 Tf
0.8265 0 TD
(y)Tj
/F12 1 Tf
4.9813 0 0 4.9813 150.127 296.1215 Tm
(0)Tj
/F2 1 Tf
10.9091 0 0 10.9091 154.0154 310.2645 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 163.5985 308.6286 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 168.1634 310.2645 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(n)Tj
/F2 1 Tf
0.6002 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 205.1068 308.6286 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 209.6717 310.2645 Tm
(\(1\)\))Tj
/F9 1 Tf
1.6667 1.15 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F2 1 Tf
-6.4909 -2.651 TD
(=)Tj
/F8 1 Tf
1.0555 0 TD
(y)Tj
/F1 1 Tf
7.9701 0 0 7.9701 173.9078 279.1621 Tm
(0)Tj
/F7 1 Tf
10.9091 0 0 10.9091 178.6401 280.798 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 200.8398 279.1621 Tm
(0)Tj
/F2 1 Tf
10.9091 0 0 10.9091 205.5721 280.798 Tm
(\))Tj
/F7 1 Tf
0.3889 0 TD
0.2778 Tc
(|)Tj
/F8 1 Tf
1.6111 0 TD
0 Tc
(y)Tj
/F1 1 Tf
7.9701 0 0 7.9701 232.7382 279.1621 Tm
(0)Tj
/F2 1 Tf
10.9091 0 0 10.9091 239.2887 280.798 Tm
(exp\()Tj
/F7 1 Tf
1.9167 0 TD
()Tj
/F2 1 Tf
0.7778 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 279.1701 279.1621 Tm
(0)Tj
/F2 1 Tf
10.9091 0 0 10.9091 286.3273 280.798 Tm
[(+)-222.2(1\)\(log)-13.8(\()]TJ
/F8 1 Tf
3.9584 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 335.7539 279.1621 Tm
(0)Tj
/F2 1 Tf
10.9091 0 0 10.9091 342.9101 280.798 Tm
[(+)-222.2(1)0(\))-222.2(+)-222.2(log)-180.6(log)-13.8(\()]TJ
/F8 1 Tf
6.25 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 417.337 279.1621 Tm
(0)Tj
/F2 1 Tf
10.9091 0 0 10.9091 424.4942 280.798 Tm
0.2221 Tc
[(+2)222.1(\))]TJ
/F7 1 Tf
2.111 0 TD
0 Tc
()Tj
/F8 1 Tf
1 0 TD
(C)Tj
/F1 1 Tf
7.9701 0 0 7.9701 466.2297 279.1621 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 470.9619 280.798 Tm
(\)\))Tj
-32.2816 -1.9107 TD
[(for)-333.3(some)-333.3(absolute)-333.2(constan)27.8(t)]TJ
/F8 1 Tf
12.0195 0 TD
(C)Tj
/F1 1 Tf
7.9701 0 0 7.9701 257.7176 258.3183 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 262.4498 259.9542 Tm
(.)Tj
-11.1588 -1.2694 TD
[(F)83.4(rom)-223(the)-223(last)-223(t)27.7(w)27.8(o)-223(displa)27.8(y)27.8(e)0(d)-223(equations)-222.9(w)27.8(e)-223(conclude)-222.9(that)-223(if)]TJ
/F8 1 Tf
24.5415 0 TD
(D)Tj
/F2 1 Tf
1.0787 0 TD
[(is)-223(large)-223(enough)]TJ
-27.6293 -1.2694 TD
[(\(in)-333.3(terms)-333.3(of)]TJ
/F8 1 Tf
5.4805 0 TD
(C)Tj
/F1 1 Tf
7.9701 0 0 7.9701 186.3841 230.6221 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 194.7527 232.258 Tm
(and)Tj
/F8 1 Tf
1.9445 0 TD
(C)Tj
/F1 1 Tf
7.9701 0 0 7.9701 223.7619 230.6221 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 228.4941 232.258 Tm
[(\))-333.4(then)]TJ
/F9 1 Tf
-7.9745 -1.0826 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 133.0401 196.9375 Tm
(y)Tj
/F12 1 Tf
4.9813 0 0 4.9813 137.2264 195.9412 Tm
(0)Tj
/F15 1 Tf
7.9701 0 0 7.9701 141.1138 196.9375 Tm
()Tj
/F14 1 Tf
0.8265 0 TD
(n)Tj
/F15 1 Tf
0.6446 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(w)Tj
/F2 1 Tf
10.9091 0 0 10.9091 165.7166 210.0852 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 175.2996 208.4483 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 179.8645 210.0852 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(n)Tj
/F2 1 Tf
0.6002 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 216.808 208.4483 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 221.3729 210.0852 Tm
(\(1\)\))Tj
/F7 1 Tf
1.9445 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(w)Tj
/F2 1 Tf
0.9095 0 TD
(exp)Tj
/F7 1 Tf
1.5278 0 TD
({)Tj
/F8 1 Tf
1.2778 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
[(\(log)-13.9(\()]TJ
/F8 1 Tf
2.0694 0 TD
(u)Tj
/F2 1 Tf
0.7947 0 TD
[(+)-222.2(1)0(\))-222.1(+)-222.2(log)-180.5(log)-13.9(\()]TJ
/F8 1 Tf
6.25 0 TD
(u)Tj
/F2 1 Tf
0.7947 0 TD
0.2222 Tc
[(+2)222.2(\)+)]TJ
/F8 1 Tf
3.1111 0 TD
0 Tc
(O)Tj
/F2 1 Tf
0.7906 0 TD
(\(1\)\))Tj
/F7 1 Tf
1.6667 0 TD
(})Tj
/F2 1 Tf
-32.1672 -2.8742 TD
[(so)-394.4(that)-394.3(the)-394.4(lo)27.8(w)27.8(e)0(r)-394.4(b)-27.8(ound)-394.3(in)-394.4(the)-394.3(corollary)-394.4(follo)27.8(ws)-394.4(with)]TJ
/F8 1 Tf
24.0188 0 TD
(v)Tj
/F1 1 Tf
7.9701 0 0 7.9701 386.1101 177.0929 Tm
(+)Tj
/F2 1 Tf
10.9091 0 0 10.9091 397.335 178.7297 Tm
(=)Tj
/F8 1 Tf
1.1573 0 TD
(w)Tj
/F7 1 Tf
1.0057 0 TD
()Tj
/F8 1 Tf
1.0407 0 TD
0.0315 Tc
(ky)Tj
/F2 1 Tf
1.4726 0 TD
0 Tc
[(for)-394.4(some)]TJ
-30.2088 -1.2694 TD
(1)Tj
/F7 1 Tf
0.7778 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(k)Tj
/F7 1 Tf
0.8299 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
0.0269 Tc
[(w/)26.9(y)]TJ
/F2 1 Tf
1.769 0 TD
0 Tc
[(].)-444.4(The)-333.4(pro)-27.7(of)-333.3(of)-333.3(the)-333.3(upp)-27.7(er)-333.4(b)-27.7(ound)-333.3(in)-333.3(the)-333.3(corollary)-333.3(is)-333.3(similar.)]TJ
-3.7563 -2.1826 TD
[(W)83.3(e)-493.7(no)27.8(w)-493.8(e)0(m)27.8(bark)-493.7(on)-493.7(the)-493.7(pro)-27.7(of)-493.8(of)-493.7(Theorem)-493.6(3.1.)-925.6(W)83.3(e)-493.7(will)-493.6(write,)-533.8(b)-27.8(elo)27.8(w,)]TJ
/F8 1 Tf
-2.0091 -1.2694 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 124.1405 125.5861 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 128.7054 127.2229 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(n)Tj
/F2 1 Tf
0.6003 0 TD
0.5518 Tc
(\)=)Tj
/F9 1 Tf
2.2701 0.75 TD
0 Tc
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 175.7768 123.9971 Tm
(d)Tj
/F15 1 Tf
0.5466 0 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
(n)Tj
/F8 1 Tf
10.9091 0 0 10.9091 189.9407 127.2229 Tm
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 195.1442 125.5861 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 199.7091 127.2229 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
[(\))-497.8(for)-497.6(a)-497.7(m)27.8(ultiplicativ)27.9(e)-497.7(function)]TJ
/F8 1 Tf
14.1303 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 368.9813 125.5861 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 373.5462 127.2229 Tm
[(,)-538.8(the)-497.7(co)-27.7(ecien)27.9(ts)-497.8(of)-497.7(the)]TJ
ET
endstream
endobj
76 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F11 14 0 R
/F12 15 0 R
/F14 22 0 R
/F15 23 0 R
/F16 24 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
78 0 obj
<<
/Length 21475
>>
stream
BT
/F5 1 Tf
7.9701 0 0 7.9701 162.3741 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
[(AN)-377.7(UNCER)87.7(T)87.5(AINTY)-377.5(PRINCIPLE)-377.6(F)29.2(O)0(R)-377.8(ARITHMETIC)-377.4(SEQUENCES)]TJ
/F2 1 Tf
10.9091 0 0 10.9091 478.5006 726.9741 Tm
(19)Tj
-32.9726 -2.3745 TD
[(Diric)27.8(hlet)-342.1(series)]TJ
/F8 1 Tf
6.8259 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 201.8411 699.4351 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 206.406 701.071 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(s)Tj
/F2 1 Tf
0.4688 0 TD
[(\).)-470.8(Note)-342.1(that)]TJ
/F8 1 Tf
5.7384 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 283.5656 699.4351 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 288.1305 701.071 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 297.861 705.0302 Tm
(k)Tj
/F2 1 Tf
10.9091 0 0 10.9091 302.9808 701.071 Tm
0.2924 Tc
[(\)=0)-49.7(f)292.4(o)292.4(r)]TJ
/F8 1 Tf
4.1329 0 TD
0 Tc
(p)Tj
/F17 1 Tf
0.7955 0 TD
()Tj
/F8 1 Tf
0.5702 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
[(,)-344.3(and)-342(if)]TJ
/F8 1 Tf
3.5007 0 TD
(p)Tj
/F7 1 Tf
0.5031 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(q)Tj
/F2 1 Tf
0.8244 0 TD
(then)Tj
/F8 1 Tf
2.2866 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 454.0772 699.4351 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 458.6421 701.071 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 468.3726 705.0302 Tm
(k)Tj
/F2 1 Tf
10.9091 0 0 10.9091 473.4924 701.071 Tm
0.2924 Tc
(\)=)Tj
/F8 1 Tf
-32.5135 -1.2694 TD
0 Tc
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 124.1405 685.587 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 128.7054 687.2229 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 138.4369 691.1821 Tm
(k)Tj
/F2 1 Tf
10.9091 0 0 10.9091 143.5567 687.2229 Tm
(\))Tj
/F7 1 Tf
0.7074 0 TD
()Tj
/F8 1 Tf
1.0963 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 168.5748 685.587 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 173.1397 687.2229 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 182.8702 691.1821 Tm
(k)Tj
/F15 1 Tf
0.5799 0 TD
()Tj
/F1 1 Tf
0.8265 0 TD
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 198.8115 687.2229 Tm
[(\).)-877.9(Clearly)]TJ
/F8 1 Tf
5.1641 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 263.724 685.587 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 268.2889 687.2229 Tm
[(\(1\))-518.6(=)]TJ
/F9 1 Tf
3.0928 0.75 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 313.5442 692.2501 Tm
()Tj
/F14 1 Tf
0 -1.0355 TD
(d)Tj
/F1 1 Tf
0.5466 0 TD
0.0001 Tc
(=1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 331.0386 687.2229 Tm
0 Tc
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 336.2411 685.587 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 340.806 687.2229 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/d)Tj
/F2 1 Tf
1.0205 0 TD
[(.)-877.9(Let)-477.8([)]TJ
/F8 1 Tf
3.3697 0 TD
(t)Tj
/F2 1 Tf
0.3611 0 TD
[(])-477.8(and)]TJ
/F7 1 Tf
2.8446 0 TD
({)Tj
/F8 1 Tf
0.5 0 TD
(t)Tj
/F7 1 Tf
0.3611 0 TD
(})Tj
/F2 1 Tf
0.9779 0 TD
(denote)Tj
-31.0837 -1.2694 TD
[(resp)-27.7(ectiv)27.8(ely)-333.3(the)-333.3(in)27.8(teger)-333.3(and)-333.2(fractional)-333.3(part)-333.3(of)]TJ
/F8 1 Tf
20.2111 0 TD
(t)Tj
/F2 1 Tf
0.3611 0 TD
[(.)-444.4(Then)]TJ
/F8 1 Tf
-17.2542 -2.7174 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
0.2778 Tc
(\)=)Tj
2.0838 0.6765 TD
0 Tc
(1)Tj
ET
194.13 646.726 10.954 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 194.1479 636.2471 Tm
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 199.7011 639.3983 Tm
(u)Tj
/F9 1 Tf
10.9091 0 0 10.9091 210.5933 654.0941 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 208.1156 630.5833 Tm
(n)Tj
/F15 1 Tf
0.6448 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F10 1 Tf
4.9813 0 0 4.9813 224.1166 632.8528 Tm
(u)Tj
/F9 1 Tf
10.9091 0 0 10.9091 230.6471 654.0941 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 232.6017 629.9348 Tm
(d)Tj
/F15 1 Tf
0.5468 0 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
(n)Tj
/F8 1 Tf
10.9091 0 0 10.9091 248.2232 643.73 Tm
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 253.4267 642.0941 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 257.9915 643.73 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
1.1096 0.6765 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 295.2657 655.0695 Tm
(u)Tj
/F2 1 Tf
10.9091 0 0 10.9091 300.6675 651.1104 Tm
(])Tj
ET
286.65 646.726 17.015 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 289.7136 636.2471 Tm
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 295.2657 639.3983 Tm
(u)Tj
/F8 1 Tf
10.9091 0 0 10.9091 304.8927 643.73 Tm
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 313.4705 642.0941 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 318.0354 643.73 Tm
[(\(1\))-277.7(=)]TJ
2.9728 0.6765 TD
(1)Tj
ET
347.73 646.726 10.954 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 347.7151 636.2471 Tm
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 353.2682 639.3983 Tm
(u)Tj
/F15 1 Tf
1.5131 2.2545 TD
()Tj
/F9 1 Tf
10.9091 0 0 10.9091 361.6837 654.0941 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 361.9736 630.3771 Tm
(d)Tj
/F1 1 Tf
0.5466 0 TD
(=1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 379.2588 643.73 Tm
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 384.4623 642.0941 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 389.0272 643.73 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5206 0 TD
(\))Tj
/F9 1 Tf
0.3889 1.1101 TD
( )Tj
/F8 1 Tf
1.1791 -0.4335 TD
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 421.605 655.0695 Tm
(u)Tj
ET
416.01 646.726 10.954 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 418.6899 636.2471 Tm
(d)Tj
/F9 1 Tf
0.872 1.796 TD
()Tj
/F7 1 Tf
0.6944 -1.1101 TD
()Tj
/F2 1 Tf
1.1096 0.6765 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 456.4653 655.0695 Tm
(u)Tj
/F2 1 Tf
10.9091 0 0 10.9091 461.867 651.1104 Tm
(])Tj
ET
447.93 646.726 17.015 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 453.5512 636.2471 Tm
(d)Tj
/F9 1 Tf
1.1496 1.796 TD
(\012)Tj
/F8 1 Tf
0.5973 -1.1101 TD
(.)Tj
/F2 1 Tf
-32.4324 0 TD
(\(3.1\))Tj
0 -3.1556 TD
[(W)83.3(e)-333.3(b)-27.8(egin)-333.3(b)27.8(y)-333.3(establishing)-333.2(the)-333.4(upp)-27.7(er)-333.3(b)-27.8(ound)-333.3(for)]TJ
/F7 1 Tf
20.4611 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
(|)Tj
/F2 1 Tf
0.611 0 TD
[(in)-333.3(Theorem)-333.3(3.1.)]TJ
/F5 1 Tf
-21.4869 -2.1827 TD
[(Pr)22.3(oposition)-377.7(3.6)]TJ
/F2 1 Tf
7.8555 0 TD
(.)Tj
/F16 1 Tf
1.1667 0 TD
[(In)-357.7(the)-357.7(r)51.1(ange)]TJ
/F2 1 Tf
5.6883 0 TD
(1)Tj
/F7 1 Tf
0.7777 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
()Tj
/F7 1 Tf
0.7613 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 342.2107 589.7893 Tm
(2)Tj
ET
342.21 588.406 4.234 -0.436 re
f
BT
7.9701 0 0 7.9701 342.2107 581.7325 Tm
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 349.4586 585.4944 Tm
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(,)Tj
/F7 1 Tf
-13.5811 -2.3532 TD
(|)Tj
/F8 1 Tf
0.2777 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
0.2777 Tc
(|)Tj
/F2 1 Tf
1.611 0 TD
0 Tc
(exp\()Tj
/F8 1 Tf
1.9167 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1 0 TD
0.046 Tc
(u)Tj
/F2 1 Tf
1.2782 0 TD
0.2222 Tc
(+5)Tj
/F8 1 Tf
1.5 0 TD
0 Tc
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F9 1 Tf
0.3889 1.1099 TD
(\012)Tj
/F8 1 Tf
0.5973 -1.11 TD
(,)Tj
/F16 1 Tf
-24.1649 -2.2029 TD
[(and)-357.7(also)]TJ
/F9 1 Tf
8.1243 -0.2301 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 218.3372 530.1251 Tm
()Tj
/F1 1 Tf
-0.6084 -2.7084 TD
(0)Tj
/F8 1 Tf
10.9091 0 0 10.9091 229.1218 518.4319 Tm
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 234.2007 522.936 Tm
0.0391 Tc
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 243.6354 518.4319 Tm
0 Tc
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(du)Tj
/F7 1 Tf
1.3707 0 TD
()Tj
/F2 1 Tf
1.1652 0.6764 TD
(3)Tj
ET
300.81 521.326 5.455 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 300.8618 510.9489 Tm
()Tj
/F2 1 Tf
0.768 0.6859 TD
(exp\()Tj
/F8 1 Tf
1.9166 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
0.2222 Tc
(\)+5)Tj
/F8 1 Tf
2.1111 0 TD
0 Tc
(J)Tj
/F2 1 Tf
0.6508 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\)\))Tj
/F8 1 Tf
0.7778 0 TD
(.)Tj
/F2 1 Tf
-23.5615 -3.5019 TD
[(As)-401.2(will)-401.2(b)-27.8(e)-401.2(eviden)27.9(t)-401.2(from)-401.2(the)-401.2(pro)-27.7(of,)-418.2(the)-401.2(condition)]TJ
/F8 1 Tf
21.9447 0 TD
()Tj
/F7 1 Tf
0.8744 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 403.5985 484.524 Tm
(2)Tj
ET
403.65 483.166 4.234 -0.436 re
f
BT
7.9701 0 0 7.9701 403.5985 476.4673 Tm
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 410.8463 480.2292 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.9102 0 TD
[(ma)27.8(y)-401.3(b)-27.7(e)-401.3(re-)]TJ
-29.1395 -1.2694 TD
[(placed)-405.4(b)27.8(y)]TJ
/F8 1 Tf
4.6444 0 TD
()Tj
/F7 1 Tf
0.8816 0 TD
()Tj
/F2 1 Tf
1.1758 0 TD
(\(1)Tj
/F7 1 Tf
1.1592 0 TD
()Tj
/F8 1 Tf
1.0481 0 TD
()Tj
/F2 1 Tf
0.4663 0 TD
[(\))-166.7(log)]TJ
/F8 1 Tf
2.0139 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
[(.)-660.9(The)-405.5(constan)27.9(ts)-405.5(3)-405.5(and)-405.5(5)-405.5(will)-405.5(ha)27.8(v)27.8(e)-405.5(to)-405.4(b)-27.8(e)-405.5(replaced)]TJ
-11.8983 -1.2694 TD
[(with)-333.3(appropriate)-333.2(constan)27.8(ts)-333.3(dep)-27.8(ending)-333.2(only)-333.3(on)]TJ
/F8 1 Tf
20.4889 0 TD
()Tj
/F2 1 Tf
0.4663 0 TD
(.)Tj
/F16 1 Tf
-18.9461 -2.1826 TD
[(Pr)51.2(o)51.1(o)0(f)-285.9(o)0(f)-285.9(P)0(r)51.1(o)0(p)51.2(osition)]TJ
/F2 1 Tf
8.9609 0 TD
[(3.6.)-836.7(Since)]TJ
/F7 1 Tf
4.9253 0 TD
(|)Tj
/F2 1 Tf
0.2778 0 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 303.8177 432.6815 Tm
(u)Tj
/F8 1 Tf
10.9091 0 0 10.9091 309.2184 428.7223 Tm
(/d)Tj
/F2 1 Tf
1.0205 0 TD
(])Tj
/F7 1 Tf
0.3437 0 TD
()Tj
/F2 1 Tf
0.8437 0 TD
([)Tj
/F8 1 Tf
0.2777 0 TD
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 341.8879 432.6815 Tm
(u)Tj
/F2 1 Tf
10.9091 0 0 10.9091 347.2897 428.7223 Tm
(])Tj
/F8 1 Tf
0.2777 0 TD
(/d)Tj
/F7 1 Tf
1.0205 0 TD
0.2777 Tc
(|)Tj
/F2 1 Tf
1.611 0 TD
0 Tc
(min\()Tj
/F8 1 Tf
2.0556 0 TD
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 407.0047 432.6815 Tm
(u)Tj
/F8 1 Tf
10.9091 0 0 10.9091 412.4065 428.7223 Tm
(/d,)Tj
/F2 1 Tf
1.4649 0 TD
[(1\))-255.1(w)27.7(e)-255.2(obtain,)]TJ
-28.3789 -1.2694 TD
[(from)-333.3(\(3.1\),)-333.3(that)]TJ
/F7 1 Tf
3.7611 -2.773 TD
(|)Tj
/F8 1 Tf
0.2777 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
0.2777 Tc
(|)Tj
/F9 1 Tf
1.8025 0.95 TD
0 Tc
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 203.8466 371.2708 Tm
(d)Tj
/F15 1 Tf
0.5467 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F10 1 Tf
4.9813 0 0 4.9813 219.0665 373.5403 Tm
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 226.7925 392.0638 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 235.0266 390.4279 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 239.5915 392.0638 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
(|)Tj
ET
226.77 387.526 29.992 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 236.3118 377.1407 Tm
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 241.865 380.2919 Tm
(u)Tj
/F2 1 Tf
10.9091 0 0 10.9091 260.4045 384.6236 Tm
(+)Tj
/F9 1 Tf
1.1913 0.95 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 271.3136 371.2708 Tm
(d>z)Tj
/F10 1 Tf
4.9813 0 0 4.9813 286.5325 373.5403 Tm
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 294.2596 392.0638 Tm
(|)Tj
/F8 1 Tf
0.2777 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 302.4926 390.4279 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 307.0575 392.0638 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
ET
294.21 387.526 29.992 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 306.416 377.1407 Tm
(d)Tj
/F7 1 Tf
2.0222 0.6859 TD
()Tj
/F8 1 Tf
1.0556 0 TD
(e)Tj
/F15 1 Tf
7.9701 0 0 7.9701 345.071 389.1277 Tm
()Tj
/F14 1 Tf
0.8265 0 TD
0.039 Tc
(u)Tj
/F15 1 Tf
1.8691 1.1458 TD
0 Tc
()Tj
/F9 1 Tf
10.9091 0 0 10.9091 362.9111 394.9868 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 363.2 371.2708 Tm
(d)Tj
/F1 1 Tf
0.5467 0 TD
(=1)Tj
/F7 1 Tf
10.9091 0 0 10.9091 381.6817 392.0638 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 389.9158 390.4279 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 394.4807 392.0638 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
(|)Tj
ET
381.69 387.526 29.992 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 393.8391 377.1407 Tm
(d)Tj
1.7444 0.6859 TD
(d)Tj
/F14 1 Tf
7.9701 0 0 7.9701 418.5474 389.1277 Tm
0.039 Tc
(/)Tj
/F1 1 Tf
1.2144 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 445.3499 384.6236 Tm
(;)Tj
-29.9338 -3.0001 TD
[(and)-333.3(also)-333.3(that)]TJ
/F9 1 Tf
1.9674 -0.8837 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 151.1711 339.0993 Tm
()Tj
/F1 1 Tf
-0.6083 -2.7084 TD
(0)Tj
/F8 1 Tf
10.9091 0 0 10.9091 161.9557 327.4062 Tm
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 167.0356 331.9103 Tm
0.0391 Tc
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 176.4702 327.4062 Tm
0 Tc
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(du)Tj
/F7 1 Tf
1.2596 0 TD
()Tj
/F9 1 Tf
1.0555 0.75 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 242.7148 332.4324 Tm
()Tj
/F14 1 Tf
0 -1.0354 TD
(d)Tj
/F1 1 Tf
0.5466 0 TD
(=1)Tj
/F7 1 Tf
10.9091 0 0 10.9091 260.2082 327.4062 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 268.4424 325.7693 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 273.0072 327.4062 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F9 1 Tf
0.2778 1.11 TD
( )Tj
0.7638 -0.3045 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 305.8062 333.0391 Tm
(log)Tj
/F14 1 Tf
1.5495 0 TD
(d/)Tj
/F1 1 Tf
1.2551 0 TD
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F1 1 Tf
-4.6202 -1.1875 TD
(0)Tj
/F14 1 Tf
5.5972 1.0196 TD
(e)Tj
/F10 1 Tf
4.9813 0 0 4.9813 352.2242 334.5136 Tm
0.0183 Tc
(u)Tj
ET
348.33 330.406 12.053 -0.436 re
f
BT
/F14 1 Tf
7.9701 0 0 7.9701 352.1435 323.6443 Tm
0 Tc
(d)Tj
/F8 1 Tf
10.9091 0 0 10.9091 361.5442 327.4062 Tm
(du)Tj
/F2 1 Tf
1.3152 0 TD
(+)Tj
/F9 1 Tf
1 0.8055 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 394.0732 333.0391 Tm
()Tj
/F1 1 Tf
-0.2661 -1.1875 TD
(log)Tj
/F14 1 Tf
1.5495 0 TD
(d/)Tj
/F1 1 Tf
1.255 0 TD
(log)Tj
/F14 1 Tf
1.5496 0 TD
(z)Tj
0.977 1.0196 TD
(e)Tj
/F10 1 Tf
4.9813 0 0 4.9813 438.3701 334.5136 Tm
0.0183 Tc
(u)Tj
ET
434.49 330.406 12.053 -0.436 re
f
BT
/F14 1 Tf
7.9701 0 0 7.9701 435.9741 323.6443 Tm
0 Tc
(z)Tj
/F10 1 Tf
4.9813 0 0 4.9813 440.2501 325.9128 Tm
(u)Tj
/F8 1 Tf
10.9091 0 0 10.9091 447.6902 327.4062 Tm
(du)Tj
/F9 1 Tf
1.093 1.11 TD
(\012)Tj
/F7 1 Tf
-18.3298 -3.5321 TD
()Tj
/F9 1 Tf
1.0556 1.11 TD
( )Tj
/F1 1 Tf
7.9701 0 0 7.9701 278.8782 305.2782 Tm
(1)Tj
ET
278.85 303.886 4.234 -0.436 re
f
BT
/F14 1 Tf
7.9701 0 0 7.9701 278.9788 297.2214 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 286.7318 300.9833 Tm
(+)Tj
/F1 1 Tf
7.9701 0 0 7.9701 310.3422 305.2782 Tm
(1)Tj
ET
298.89 303.886 27.246 -0.436 re
f
BT
7.9701 0 0 7.9701 298.8364 297.2214 Tm
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F15 1 Tf
0.5365 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
()Tj
/F9 1 Tf
10.9091 0 0 10.9091 327.2777 313.0919 Tm
(\012)Tj
0.7639 -0.36 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 347.1263 306.0104 Tm
()Tj
/F14 1 Tf
0 -1.0355 TD
(d)Tj
/F1 1 Tf
0.5468 0 TD
(=1)Tj
/F15 1 Tf
1.7982 1.079 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
(g)Tj
/F10 1 Tf
4.9813 0 0 4.9813 372.1983 305.3609 Tm
(q)Tj
/F1 1 Tf
7.9701 0 0 7.9701 376.1574 306.3572 Tm
(\()Tj
/F14 1 Tf
0.4132 0 TD
(d)Tj
/F1 1 Tf
0.5466 0 TD
(\))Tj
/F15 1 Tf
0.4132 0 TD
(|)Tj
ET
1 i
365.85 303.45 23.637 0.436 re
f
BT
/F14 1 Tf
7.9701 0 0 7.9701 375.4561 297.2214 Tm
(d)Tj
/F8 1 Tf
10.9091 0 0 10.9091 390.6491 300.9833 Tm
(d)Tj
/F14 1 Tf
7.9701 0 0 7.9701 396.3268 304.9415 Tm
0.0391 Tc
(/)Tj
/F1 1 Tf
1.2145 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F8 1 Tf
10.9091 0 0 10.9091 423.1293 300.9833 Tm
(.)Tj
/F2 1 Tf
-27.8969 -2.4193 TD
[(No)27.8(w,)-333.3(as)-333.4(eac)27.8(h)]TJ
/F7 1 Tf
6.0333 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(g)Tj
/F2 1 Tf
0.5128 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 202.9738 278.5504 Tm
(k)Tj
/F2 1 Tf
10.9091 0 0 10.9091 208.0937 274.5912 Tm
(\))Tj
/F7 1 Tf
0.3889 0 TD
0.2778 Tc
(|)Tj
/F2 1 Tf
1.6111 0 TD
0 Tc
[(1)-333.3(and)-333.3(as)-333.3(2)]TJ
/F1 1 Tf
7.9701 0 0 7.9701 279.0635 278.5504 Tm
(1)Tj
/F14 1 Tf
0.5312 0 TD
(/)Tj
/F1 1 Tf
0.5313 0 TD
(3)Tj
/F8 1 Tf
10.9091 0 0 10.9091 292.264 274.5912 Tm
(/)Tj
/F2 1 Tf
0.5 0 TD
(\(2)Tj
/F1 1 Tf
7.9701 0 0 7.9701 307.4152 278.5504 Tm
(1)Tj
/F14 1 Tf
0.5312 0 TD
(/)Tj
/F1 1 Tf
0.5312 0 TD
(3)Tj
/F7 1 Tf
10.9091 0 0 10.9091 323.0406 274.5912 Tm
()Tj
/F2 1 Tf
1 0 TD
(1\))Tj
/F8 1 Tf
1.1667 0 TD
(<)Tj
/F2 1 Tf
1.0555 0 TD
(5,)Tj
/F15 1 Tf
7.9701 0 0 7.9701 167.5138 257.9766 Tm
()Tj
/F9 1 Tf
10.9091 0 0 10.9091 163.8695 254.7039 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 164.1594 230.9878 Tm
(d)Tj
/F1 1 Tf
0.5466 0 TD
(=1)Tj
/F7 1 Tf
10.9091 0 0 10.9091 182.6401 251.7808 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 190.8742 250.145 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 195.4391 251.7808 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
ET
182.61 247.246 29.992 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 194.7975 236.8578 Tm
(d)Tj
1.7445 0.6859 TD
(d)Tj
/F14 1 Tf
7.9701 0 0 7.9701 219.5059 248.8448 Tm
0.0391 Tc
(/)Tj
/F1 1 Tf
1.2144 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F7 1 Tf
10.9091 0 0 10.9091 248.1265 244.3407 Tm
()Tj
/F9 1 Tf
1.0556 0.75 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 269.9447 241.1148 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
(q)Tj
/F9 1 Tf
10.9091 0 0 10.9091 282.943 256.4493 Tm
( )Tj
/F2 1 Tf
0.5973 -1.11 TD
0.2222 Tc
(1+)Tj
/F1 1 Tf
7.9701 0 0 7.9701 312.3726 249.7145 Tm
0 Tc
(1)Tj
/F15 1 Tf
0.5312 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(f)Tj
/F10 1 Tf
4.9813 0 0 4.9813 327.2618 248.7183 Tm
(q)Tj
/F1 1 Tf
7.9701 0 0 7.9701 331.2209 249.7145 Tm
(\()Tj
/F14 1 Tf
0.4131 0 TD
(p)Tj
/F1 1 Tf
0.5349 0 TD
(\))Tj
ET
309.45 247.246 35.559 -0.436 re
f
BT
/F14 1 Tf
7.9701 0 0 7.9701 309.4416 240.4084 Tm
(p)Tj
/F12 1 Tf
4.9813 0 0 4.9813 313.7046 242.677 Tm
(1)Tj
/F11 1 Tf
0.6806 0 TD
()Tj
/F10 1 Tf
1.0834 0 TD
0.0181 Tc
(/)Tj
/F12 1 Tf
1.6664 0 TD
-0.0001 Tc
(log)Tj
/F10 1 Tf
2.0266 0 TD
0 Tc
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 348.6207 244.3407 Tm
(+)Tj
/F9 1 Tf
0.9999 0.75 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 371.0446 249.3679 Tm
()Tj
/F14 1 Tf
0 -1.0355 TD
(k)Tj
/F1 1 Tf
0.5799 0 TD
(=2)Tj
4.3439 0.9436 TD
(1)Tj
ET
389.97 247.246 44.812 -0.436 re
f
BT
/F14 1 Tf
7.9701 0 0 7.9701 389.9985 240.4084 Tm
(p)Tj
/F10 1 Tf
4.9813 0 0 4.9813 394.2615 242.677 Tm
(k)Tj
/F12 1 Tf
0.7742 0 TD
-0.0001 Tc
(\(1)Tj
/F11 1 Tf
1.2224 0 TD
0 Tc
()Tj
/F10 1 Tf
1.0832 0 TD
0.0183 Tc
(/)Tj
/F12 1 Tf
1.6664 0 TD
-0.0001 Tc
(log)Tj
/F10 1 Tf
2.0266 0 TD
0 Tc
(z)Tj
/F12 1 Tf
0.7258 0 TD
(\))Tj
/F9 1 Tf
10.9091 0 0 10.9091 436.006 256.4493 Tm
(\012)Tj
/F7 1 Tf
-16.7063 -3.9493 TD
()Tj
/F9 1 Tf
1.0555 0.75 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 275.5736 210.141 Tm
(p)Tj
/F15 1 Tf
0.5348 0 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
(q)Tj
/F9 1 Tf
10.9091 0 0 10.9091 288.5709 225.4754 Tm
( )Tj
/F2 1 Tf
0.5973 -1.11 TD
0.2222 Tc
(1+)Tj
/F1 1 Tf
7.9701 0 0 7.9701 318.0005 218.7407 Tm
0 Tc
(1)Tj
/F15 1 Tf
0.5312 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(f)Tj
/F10 1 Tf
4.9813 0 0 4.9813 332.8897 217.7444 Tm
(q)Tj
/F1 1 Tf
7.9701 0 0 7.9701 336.8488 218.7407 Tm
(\()Tj
/F14 1 Tf
0.4131 0 TD
(p)Tj
/F1 1 Tf
0.5349 0 TD
(\))Tj
ET
315.09 216.286 35.559 -0.436 re
f
BT
/F14 1 Tf
7.9701 0 0 7.9701 315.0695 209.4336 Tm
(p)Tj
/F12 1 Tf
4.9813 0 0 4.9813 319.3325 211.7031 Tm
(1)Tj
/F11 1 Tf
0.6806 0 TD
()Tj
/F10 1 Tf
1.0834 0 TD
0.0181 Tc
(/)Tj
/F12 1 Tf
1.6664 0 TD
-0.0001 Tc
(log)Tj
/F10 1 Tf
2.0266 0 TD
0 Tc
(z)Tj
/F9 1 Tf
10.9091 0 0 10.9091 351.8237 225.4754 Tm
(\012 )Tj
/F2 1 Tf
1.1944 -1.11 TD
0.2223 Tc
(1+)Tj
/F1 1 Tf
7.9701 0 0 7.9701 404.8927 217.6618 Tm
0 Tc
(5)Tj
ET
384.81 215.85 44.346 0.436 re
f
BT
/F14 1 Tf
7.9701 0 0 7.9701 384.8379 209.4336 Tm
(p)Tj
/F12 1 Tf
4.9813 0 0 4.9813 389.0999 211.7031 Tm
-0.0001 Tc
(2\(1)Tj
/F11 1 Tf
1.903 0 TD
0 Tc
()Tj
/F10 1 Tf
1.0832 0 TD
0.0183 Tc
(/)Tj
/F12 1 Tf
1.6664 0 TD
-0.0001 Tc
(log)Tj
/F10 1 Tf
2.0266 0 TD
0 Tc
(z)Tj
/F12 1 Tf
0.7258 0 TD
(\))Tj
/F9 1 Tf
10.9091 0 0 10.9091 430.3781 225.4754 Tm
(\012)Tj
/F2 1 Tf
-28.5614 -3.483 TD
(since)Tj
/F8 1 Tf
2.4795 0 TD
(p)Tj
/F7 1 Tf
0.8302 0 TD
()Tj
/F2 1 Tf
1.1048 0 TD
[(2)-362.9(and)]TJ
/F8 1 Tf
2.8369 0 TD
()Tj
/F7 1 Tf
0.8106 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 219.998 191.7739 Tm
(2)Tj
ET
220.05 190.366 4.234 -0.436 re
f
BT
7.9701 0 0 7.9701 219.998 183.717 Tm
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 227.2458 187.4789 Tm
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
[(.)-533.2(The)-362.9(prop)-27.8(osition)-362.8(follo)27.8(ws)-362.9(up)-27.8(on)-362.9(taking)-362.8(logarithms.)]TJ
ET
482.61 180.369 6.775 -0.399 re
f
482.61 179.948 0.399 -5.978 re
f
488.97 179.948 0.399 -5.978 re
f
482.61 174.009 6.775 -0.399 re
f
BT
10.9091 0 0 10.9091 140.7173 148.9952 Tm
(Dene)Tj
/F8 1 Tf
9.7138 -1.5913 TD
(I)Tj
/F2 1 Tf
0.5181 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(s)Tj
/F2 1 Tf
0.4688 0 TD
0.2778 Tc
(\)=)Tj
/F9 1 Tf
1.7222 1.3612 TD
0 Tc
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 291.3903 143.3295 Tm
()Tj
/F1 1 Tf
-0.6082 -2.7084 TD
(0)Tj
/F8 1 Tf
10.9091 0 0 10.9091 302.1748 131.6353 Tm
(e)Tj
/F15 1 Tf
7.9701 0 0 7.9701 307.2548 136.1395 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
(su)Tj
/F8 1 Tf
10.9091 0 0 10.9091 323.1582 131.6353 Tm
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\))Tj
/F8 1 Tf
0.389 0 TD
(du.)Tj
ET
endstream
endobj
79 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F10 13 0 R
/F11 14 0 R
/F12 15 0 R
/F14 22 0 R
/F15 23 0 R
/F16 24 0 R
/F17 28 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
81 0 obj
<<
/Length 19256
>>
stream
BT
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
(20)Tj
/F5 1 Tf
7.9701 0 0 7.9701 202.119 726.9739 Tm
[(ANDREW)-377.5(GRANVILLE)-377.4(AND)-377.7(K)0(.)-377.7(SOUND)29.4(ARARAJAN)]TJ
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 701.071 Tm
[(F)83.4(rom)-475.7(Prop)-27.7(osition)-475.6(3.6)-475.7(it)-475.7(is)-475.6(clear)-475.7(that)]TJ
/F8 1 Tf
17.1826 0 TD
(I)Tj
/F2 1 Tf
0.5181 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(s)Tj
/F2 1 Tf
0.4687 0 TD
[(\))-475.7(con)27.8(v)27.8(erges)-475.6(absolutely)-475.6(in)-475.7(Re\()]TJ
/F8 1 Tf
13.2639 0 TD
(s)Tj
/F2 1 Tf
0.4687 0 TD
(\))Tj
/F8 1 Tf
0.9039 0 TD
(>)Tj
/F7 1 Tf
-33.1948 -1.2694 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 128.4802 691.5178 Tm
(2)Tj
ET
128.49 690.166 4.234 -0.436 re
f
BT
7.9701 0 0 7.9701 128.4802 683.4611 Tm
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 135.728 687.2229 Tm
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
[(,)-576.8(and)-528.1(th)27.7(us)-528.2(denes)-528.1(an)-528.2(analytic)-528.1(function)-528.1(in)-528.1(this)-528(region.)-1028.9(F)83.4(urther)-528.1(if)]TJ
-3.5191 -1.2694 TD
(Re\()Tj
/F8 1 Tf
1.5694 0 TD
(s)Tj
/F2 1 Tf
0.4688 0 TD
(\))Tj
/F8 1 Tf
0.6667 0 TD
(>)Tj
/F2 1 Tf
1.0556 0 TD
[(0)-333.3(then)]TJ
-3.7605 -2.0574 TD
(\(3.2\))Tj
/F8 1 Tf
0.3762 -2.2798 TD
(I)Tj
/F2 1 Tf
0.5181 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(s)Tj
/F2 1 Tf
0.4688 0 TD
0.1667 Tc
(\)=)Tj
/F9 1 Tf
1.5001 1.3611 TD
0 Tc
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 165.1846 637.7524 Tm
()Tj
/F1 1 Tf
-0.6084 -2.7084 TD
(0)Tj
/F8 1 Tf
10.9091 0 0 10.9091 177.1646 633.4386 Tm
(e)Tj
/F15 1 Tf
7.9701 0 0 7.9701 182.2436 637.3978 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
(su)Tj
ET
177.21 628.966 20.983 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 182.1788 618.5753 Tm
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 187.732 621.7275 Tm
(u)Tj
/F9 1 Tf
10.9091 0 0 10.9091 203.6384 636.4224 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 201.1606 612.9116 Tm
(n)Tj
/F15 1 Tf
0.6448 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F10 1 Tf
4.9813 0 0 4.9813 217.1616 615.1811 Tm
(u)Tj
/F2 1 Tf
10.9091 0 0 10.9091 221.874 626.0593 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 231.458 624.4224 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 236.0229 626.0593 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(n)Tj
/F2 1 Tf
0.6002 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 272.9654 624.4224 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 277.5303 626.0593 Tm
[(\(1\)\))-277.8(=)]TJ
/F15 1 Tf
7.9701 0 0 7.9701 314.0035 639.6952 Tm
()Tj
/F9 1 Tf
10.9091 0 0 10.9091 310.3582 636.4224 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 310.2575 613.1039 Tm
(n)Tj
/F1 1 Tf
0.6448 0 TD
(=1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 326.2167 626.0593 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 335.7997 624.4224 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 340.3647 626.0593 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(n)Tj
/F2 1 Tf
0.6002 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 377.3081 624.4224 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 381.873 626.0593 Tm
(\(1\)\))Tj
/F9 1 Tf
1.8333 1.3611 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 412.7821 637.7524 Tm
()Tj
/F1 1 Tf
-0.6082 -2.7084 TD
(log)Tj
/F14 1 Tf
1.5495 0 TD
(n/)Tj
/F1 1 Tf
1.353 0 TD
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F8 1 Tf
10.9091 0 0 10.9091 451.205 633.4386 Tm
(e)Tj
/F15 1 Tf
7.9701 0 0 7.9701 456.2839 637.3978 Tm
()Tj
/F14 1 Tf
0.8265 0 TD
(su)Tj
ET
451.17 628.966 20.983 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 456.2192 618.5753 Tm
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 461.7724 621.7275 Tm
(u)Tj
/F8 1 Tf
10.9091 0 0 10.9091 473.3828 626.0593 Tm
(du)Tj
/F2 1 Tf
-30.196 -3.2202 TD
(=)Tj
/F8 1 Tf
1.0541 0.6765 TD
()Tj
/F2 1 Tf
0.5112 0 TD
[(\(1)-222.2(+)]TJ
/F8 1 Tf
2.1111 0 TD
(s/)Tj
/F2 1 Tf
1.1354 0 TD
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\))Tj
ET
155.49 593.926 66.699 -0.436 re
f
BT
10.9091 0 0 10.9091 168.8658 583.4471 Tm
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F2 1 Tf
0.7313 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
(s)Tj
/F2 1 Tf
1.8061 0.6859 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 236.1843 589.2932 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 240.7492 590.9301 Tm
[(\(1)-222.2(+)]TJ
/F8 1 Tf
2.1111 0 TD
(s/)Tj
/F2 1 Tf
1.1354 0 TD
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 323.7808 589.2932 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 328.3457 590.9301 Tm
(\(1\)\))Tj
/F8 1 Tf
1.6667 0 TD
(.)Tj
/F2 1 Tf
-20.8751 -2.6243 TD
[(By)-222.2(analytic)-222.1(con)27.8(tin)27.8(uation)-222.2(this)-222.2(iden)27.8(tit)27.8(y)-222.3(con)27.9(tin)27.7(ues)-222.1(to)-222.2(hold)-222.1(for)-222.2(all)-222.2(Re\()]TJ
/F8 1 Tf
28.014 0 TD
(s)Tj
/F2 1 Tf
0.4687 0 TD
(\))Tj
/F8 1 Tf
0.6667 0 TD
(>)Tj
/F7 1 Tf
1.0556 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 457.9875 566.5963 Tm
(2)Tj
ET
458.01 565.246 4.234 -0.436 re
f
BT
7.9701 0 0 7.9701 457.9875 558.5395 Tm
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 465.2354 562.3014 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(.)Tj
/F5 1 Tf
-31.7149 -2.1827 TD
[(Pr)22.3(oposition)-377.7(3.7)]TJ
/F2 1 Tf
7.8555 0 TD
(.)Tj
/F16 1 Tf
1.1667 0 TD
-0.0767 Tc
[(Fo)-76.7(r)]TJ
/F2 1 Tf
1.8667 0 TD
0 Tc
(1)Tj
/F7 1 Tf
0.7777 0 TD
()Tj
/F8 1 Tf
1.0556 0 TD
()Tj
/F7 1 Tf
0.7613 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 300.5201 542.7856 Tm
(2)Tj
ET
300.57 541.486 4.234 -0.436 re
f
BT
7.9701 0 0 7.9701 300.5201 534.7288 Tm
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 307.7679 538.4907 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F16 1 Tf
0.8668 0 TD
(with)Tj
/F8 1 Tf
2.1722 0 TD
(z)Tj
/F16 1 Tf
0.8668 0 TD
[(suciently)-357.7(lar)51.2(ge)]TJ
/F2 1 Tf
7.1044 0 TD
(,)Tj
/F9 1 Tf
-26.783 -1.2496 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 162.5186 521.7036 Tm
()Tj
/F1 1 Tf
-0.6084 -2.7084 TD
(0)Tj
/F8 1 Tf
10.9091 0 0 10.9091 173.3031 510.0105 Tm
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 178.3831 514.5146 Tm
0.039 Tc
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 187.8177 510.0105 Tm
0 Tc
(|)Tj
/F8 1 Tf
0.2777 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(du)Tj
/F7 1 Tf
1.3707 0 TD
()Tj
/F8 1 Tf
2.513 0.6765 TD
()Tj
ET
245.01 512.926 34.682 -0.436 re
f
BT
10.9091 0 0 10.9091 244.9534 502.5275 Tm
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 250.2287 505.6787 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 257.3848 502.5275 Tm
(+)Tj
/F8 1 Tf
1 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 274.9041 505.6787 Tm
(2)Tj
/F9 1 Tf
10.9091 0 0 10.9091 280.8319 522.1191 Tm
( )Tj
/F2 1 Tf
0.7638 -1.11 TD
(exp)Tj
/F7 1 Tf
1.6944 0 TD
({)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 364.6745 508.3736 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 369.4067 510.0105 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.611 0 TD
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
0.2223 Tc
(})Tj
/F2 1 Tf
1.7223 0 TD
0 Tc
(1)Tj
/F9 1 Tf
0.5 1.11 TD
(\012)Tj
/F8 1 Tf
0.5972 -1.11 TD
(.)Tj
/F16 1 Tf
-28.6781 -3.4494 TD
[(Pr)51.2(o)51.1(o)0(f)]TJ
/F2 1 Tf
2.3267 0 TD
[(.)-888.9(T)83.3(aking)]TJ
/F8 1 Tf
4.5 0 TD
(s)Tj
/F2 1 Tf
0.7465 0 TD
(=)Tj
/F7 1 Tf
1.0556 0 TD
()Tj
/F2 1 Tf
0.7777 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.7057 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
0.0001 Tc
(i)Tj
/F2 1 Tf
0.9504 0 TD
0 Tc
[(\))-333.4(i)0(n)-333.3(\(3.2\))-333.3(w)27.8(e)-333.4(deduce)-333.3(that,)-333.3(since)]TJ
/F7 1 Tf
13.9778 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 440.6446 470.7448 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 445.2095 472.3806 Tm
(\(1\))Tj
/F7 1 Tf
1.2778 0 TD
0.2777 Tc
(|)Tj
/F2 1 Tf
1.6111 0 TD
0 Tc
(1,)Tj
/F9 1 Tf
-30.5969 -1.2368 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 153.8501 455.733 Tm
()Tj
/F1 1 Tf
-0.6084 -2.7084 TD
(0)Tj
/F8 1 Tf
10.9091 0 0 10.9091 164.6346 444.0399 Tm
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 169.7146 448.5439 Tm
0.039 Tc
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 179.1492 444.0399 Tm
0 Tc
(|)Tj
/F8 1 Tf
0.2777 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(du)Tj
/F7 1 Tf
1.3707 0 TD
0.2777 Tc
(|)Tj
/F8 1 Tf
1.3333 0 TD
0 Tc
(I)Tj
/F2 1 Tf
0.5181 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(s)Tj
/F2 1 Tf
0.4688 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
0.2777 Tc
(|)Tj
1.7207 0.6764 TD
0 Tc
(|)Tj
/F8 1 Tf
0.2777 0 TD
()Tj
/F2 1 Tf
0.5113 0 TD
[(\(1)-222.2(+)]TJ
/F8 1 Tf
2.1111 0 TD
(s/)Tj
/F2 1 Tf
1.1354 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
ET
276.09 446.926 72.759 -0.436 re
f
BT
10.9091 0 0 10.9091 289.5363 436.5569 Tm
(|)Tj
/F8 1 Tf
0.2777 0 TD
(s)Tj
/F2 1 Tf
0.691 0 TD
[(+)-222.1(log)]TJ
/F8 1 Tf
2.4583 0 TD
(z)Tj
/F7 1 Tf
0.509 0 TD
(|)Tj
/F9 1 Tf
1.6153 1.7959 TD
( )Tj
/F7 1 Tf
0.5972 -1.11 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 368.2192 442.403 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 372.7841 444.0399 Tm
[(\(1)-222.2(+)]TJ
/F8 1 Tf
2.1111 0 TD
(s/)Tj
/F2 1 Tf
1.1354 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
0.2222 Tc
(|)Tj
/F2 1 Tf
1.5 0 TD
0 Tc
(1)Tj
/F9 1 Tf
0.5 1.11 TD
(\012)Tj
/F8 1 Tf
0.5972 -1.11 TD
(.)Tj
/F2 1 Tf
-31.4818 -2.649 TD
[(F)83.4(rom)-304.7(the)-304.8(form)27.9(ula)]TJ
/F8 1 Tf
7.9337 0 TD
()Tj
/F2 1 Tf
0.5112 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
0.2778 Tc
(\)=)Tj
/F8 1 Tf
1.7222 0 TD
0.0269 Tc
(w/)Tj
/F2 1 Tf
1.2428 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F7 1 Tf
0.9079 0 TD
()Tj
/F2 1 Tf
0.9427 0 TD
(1\))Tj
/F7 1 Tf
1.054 0 TD
()Tj
/F8 1 Tf
0.9428 0 TD
(w)Tj
/F9 1 Tf
0.9095 0.8056 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 319.0276 420.7741 Tm
()Tj
/F1 1 Tf
-0.2661 -1.1875 TD
(1)Tj
/F7 1 Tf
10.9091 0 0 10.9091 327.994 415.1412 Tm
({)Tj
/F8 1 Tf
0.5 0 TD
(x)Tj
/F7 1 Tf
0.5715 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(x)Tj
/F15 1 Tf
7.9701 0 0 7.9701 351.3724 419.1003 Tm
()Tj
/F1 1 Tf
0.8265 0 TD
(1)Tj
/F15 1 Tf
0.5312 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(w)Tj
/F8 1 Tf
10.9091 0 0 10.9091 375.5696 415.1412 Tm
(dx)Tj
/F2 1 Tf
1.092 0 TD
[(,)-310.5(whic)27.9(h)-304.8(i)0(s)-304.8(v)55.6(alid)-304.7(for)-304.8(all)]TJ
-24.6292 -1.2694 TD
(Re\()Tj
/F8 1 Tf
1.5694 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\))Tj
/F8 1 Tf
0.6667 0 TD
(>)Tj
/F2 1 Tf
1.0556 0 TD
[(0,)-333.3(w)27.7(e)-333.3(glean)-333.3(that)]TJ
/F7 1 Tf
-3.8226 -1.8794 TD
(|)Tj
/F8 1 Tf
0.2777 0 TD
()Tj
/F2 1 Tf
0.5113 0 TD
[(\(1)-222.2(+)]TJ
/F8 1 Tf
2.1111 0 TD
(s/)Tj
/F2 1 Tf
1.1353 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
ET
121.17 376.366 72.759 -0.436 re
f
BT
10.9091 0 0 10.9091 134.5066 365.9278 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(s)Tj
/F2 1 Tf
0.691 0 TD
[(+)-222.2(log)]TJ
/F8 1 Tf
2.4583 0 TD
(z)Tj
/F7 1 Tf
0.509 0 TD
(|)Tj
1.8931 0.6859 TD
()Tj
/F9 1 Tf
1.0555 1.15 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F2 1 Tf
0.3333 0.05 TD
(Re)Tj
/F9 1 Tf
1.1805 1.11 TD
( )Tj
/F7 1 Tf
1.7848 -0.4335 TD
()Tj
/F2 1 Tf
0.7778 0 TD
(1)Tj
ET
233.85 376.366 37.461 -0.436 re
f
BT
10.9091 0 0 10.9091 233.8381 365.9278 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.7058 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
(i)Tj
/F2 1 Tf
0.9504 0 TD
(\))Tj
/F7 1 Tf
0.7206 0.6859 TD
()Tj
/F2 1 Tf
1.8433 0.6765 TD
(1)Tj
ET
287.01 376.366 21.462 -0.436 re
f
BT
10.9091 0 0 10.9091 287.0227 365.9278 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.7853 2.047 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 322.408 385.1048 Tm
()Tj
/F1 1 Tf
-0.6084 -2.7084 TD
(1)Tj
/F7 1 Tf
10.9091 0 0 10.9091 331.3744 373.4107 Tm
({)Tj
/F8 1 Tf
0.5 0 TD
(x)Tj
/F7 1 Tf
0.5715 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(x)Tj
/F15 1 Tf
7.9701 0 0 7.9701 354.7527 377.9148 Tm
()Tj
/F1 1 Tf
0.8265 0 TD
(2+)Tj
/F14 1 Tf
1.3576 0 TD
0.0391 Tc
(/)Tj
/F1 1 Tf
1.2144 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 400.7811 373.4107 Tm
(cos)Tj
/F9 1 Tf
1.5056 1.11 TD
( )Tj
/F8 1 Tf
0.7068 -0.4335 TD
()Tj
/F2 1 Tf
0.7726 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(x)Tj
ET
424.89 376.366 30.572 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 429.4705 365.9278 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
1.0362 1.796 TD
(\012)Tj
/F8 1 Tf
0.5973 -1.11 TD
(dx)Tj
/F9 1 Tf
1.092 1.11 TD
(\012)Tj
0.5973 0.04 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F8 1 Tf
0.3333 0.05 TD
(.)Tj
/F2 1 Tf
-33.5925 -2.6798 TD
[(F)83.4(o)0(r)-379.9(large)]TJ
/F8 1 Tf
4.3348 0 TD
(z)Tj
/F2 1 Tf
0.889 0 TD
(and)Tj
/F8 1 Tf
1.991 0 TD
()Tj
/F7 1 Tf
0.8389 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 220.2142 348.4712 Tm
(2)Tj
ET
220.17 347.086 4.234 -0.436 re
f
BT
7.9701 0 0 7.9701 220.2142 340.4144 Tm
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 227.462 344.1763 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.8889 0 TD
[(w)27.7(e)-379.8(see)-379.8(easily)-379.9(that)-379.9(the)-379.9(in)27.8(tegral)-379.8(ab)-27.7(o)27.8(v)27.8(e)-379.9(is)-379.8(p)-27.8(ositiv)27.8(e,)]TJ
/F1 1 Tf
7.9701 0 0 7.9701 484.6775 348.1345 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 330.3283 Tm
[(and)-333.3(so)-333.3(w)27.8(e)-333.4(deduce)-333.3(that)]TJ
/F7 1 Tf
4.156 -2.0574 TD
(|)Tj
/F8 1 Tf
0.2777 0 TD
()Tj
/F2 1 Tf
0.5113 0 TD
[(\(1)-222.2(+)]TJ
/F8 1 Tf
2.1111 0 TD
(s/)Tj
/F2 1 Tf
1.1353 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(/)Tj
/F7 1 Tf
0.5 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(s)Tj
/F2 1 Tf
0.691 0 TD
[(+)-222.2(log)]TJ
/F8 1 Tf
2.4583 0 TD
(z)Tj
/F7 1 Tf
0.509 0 TD
0.2778 Tc
(|)Tj
/F2 1 Tf
1.6111 0 TD
0 Tc
(Re\(1)Tj
/F8 1 Tf
2.0694 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.7057 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
0.0001 Tc
(i)Tj
/F2 1 Tf
0.9504 0 TD
0 Tc
[(\)\))-277.8(=)]TJ
/F8 1 Tf
2.1111 0 TD
0.046 Tc
(/)Tj
/F2 1 Tf
0.9836 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 407.3913 312.3875 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 414.5485 307.8834 Tm
(+)Tj
/F8 1 Tf
1 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 432.0667 312.3875 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 436.799 307.8834 Tm
(\))Tj
/F8 1 Tf
0.389 0 TD
(.)Tj
/F2 1 Tf
-27.5298 -2.0574 TD
[(Next)-479.2(w)27.8(e)-479.3(giv)27.8(e)-479.3(a)-479.3(l)0(o)27.8(w)27.7(er)-479.2(b)-27.8(ound)-479.2(for)]TJ
/F7 1 Tf
15.0551 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 316.5619 283.8018 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 321.1268 285.4386 Tm
[(\(1)-319.5(+)]TJ
/F8 1 Tf
2.3058 0 TD
(s/)Tj
/F2 1 Tf
1.1353 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F2 1 Tf
0.2778 0 TD
[(.)-882.3(W)83.3(e)-479.3(claim)-479.2(that)-479.2(for)]TJ
/F8 1 Tf
-24.6218 -1.2694 TD
(z)Tj
/F7 1 Tf
0.7868 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(101)Tj
/F1 1 Tf
7.9701 0 0 7.9701 155.2618 275.5487 Tm
(6)Tj
/F2 1 Tf
10.9091 0 0 10.9091 163.6304 271.5905 Tm
[(and)-333.3(for)-333.3(all)-333.4(primes)]TJ
/F8 1 Tf
8.0944 0 TD
(p)Tj
/F9 1 Tf
-11.6349 -2.8003 TD
()Tj
0 -0.6001 TD
()Tj
T*
()Tj
/F2 1 Tf
0.3333 0.0501 TD
0.2222 Tc
(1+)Tj
/F8 1 Tf
2.7079 0.682 TD
0 Tc
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 163.5248 234.3003 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 168.0897 235.9362 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
148.65 231.406 42.995 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 148.6266 220.5519 Tm
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 154.1151 223.718 Tm
(1+)Tj
/F14 1 Tf
1.3576 0 TD
(s/)Tj
/F1 1 Tf
1.1998 0 TD
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 195.2408 228.4961 Tm
(+)Tj
/F8 1 Tf
2.2647 0.682 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 225.2882 234.3003 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 229.8531 235.9362 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 239.5836 239.8954 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 244.3158 235.9362 Tm
(\))Tj
ET
207.33 231.406 53.815 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 207.3455 220.5519 Tm
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 212.8339 223.718 Tm
(2\(1+)Tj
/F14 1 Tf
2.3021 0 TD
(s/)Tj
/F1 1 Tf
1.1996 0 TD
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F1 1 Tf
0.5365 0 TD
(\))Tj
/F2 1 Tf
10.9091 0 0 10.9091 264.7801 228.4961 Tm
(+)Tj
/F8 1 Tf
1 0 TD
0.1667 Tc
(...)Tj
/F9 1 Tf
1.3333 1.15 TD
0 Tc
()Tj
0 -0.6001 TD
()Tj
T*
()Tj
/F7 1 Tf
0.6111 0.0501 TD
()Tj
/F9 1 Tf
1.0556 1.15 TD
()Tj
0 -0.6001 TD
()Tj
T*
()Tj
/F2 1 Tf
0.3333 0.0501 TD
0.2222 Tc
(1+)Tj
/F8 1 Tf
2.7079 0.682 TD
0 Tc
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 346.934 234.3003 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 351.4989 235.9362 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
332.01 231.406 42.995 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 332.0359 220.5519 Tm
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 337.5243 223.718 Tm
(1+)Tj
/F14 1 Tf
1.3576 0 TD
(s/)Tj
/F1 1 Tf
1.1997 0 TD
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F9 1 Tf
10.9091 0 0 10.9091 376.2262 241.0421 Tm
()Tj
0 -0.6001 TD
()Tj
T*
()Tj
0.3333 1.1601 TD
( )Tj
/F2 1 Tf
0.5972 -1.11 TD
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F2 1 Tf
1.1096 0.6765 TD
(103)Tj
ET
406.41 231.406 16.364 -0.436 re
f
BT
10.9091 0 0 10.9091 406.3602 221.0132 Tm
(100)Tj
3.6393 1.3625 TD
(1)Tj
ET
425.13 231.406 47.346 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 425.1148 220.5519 Tm
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 430.6042 223.718 Tm
(2)Tj
/F15 1 Tf
0.5312 0 TD
()Tj
/F1 1 Tf
0.8264 0 TD
(2)Tj
/F14 1 Tf
0.5312 0 TD
0.0391 Tc
(/)Tj
/F1 1 Tf
1.2144 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F9 1 Tf
10.9091 0 0 10.9091 473.6568 240.6058 Tm
(\012)Tj
/F8 1 Tf
0.5973 -1.11 TD
(.)Tj
/F2 1 Tf
-33.1258 1.9879 TD
(\(3.3\))Tj
0 -4.6237 TD
(When)Tj
/F8 1 Tf
2.9167 0 TD
0.2777 Tc
(p>)Tj
/F2 1 Tf
1.8364 0 TD
0 Tc
(101)Tj
/F1 1 Tf
7.9701 0 0 7.9701 187.0157 203.7011 Tm
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 195.3843 199.742 Tm
[(w)27.8(e)-333.4(simply)-333.3(use)-333.3(that)-333.3(the)-333.3(left)-333.3(side)-333.3(of)-333.2(\(3.3\))-333.3(exceeds)]TJ
/F7 1 Tf
1.1058 -2.7939 TD
(|)Tj
/F2 1 Tf
0.2778 0 TD
0.2222 Tc
(1+)Tj
/F8 1 Tf
1.7222 0 TD
0 Tc
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 234.6062 167.6274 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 239.1711 169.2632 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5032 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 264.0877 173.7673 Tm
(1+)Tj
/F14 1 Tf
1.3576 0 TD
(s/)Tj
/F1 1 Tf
1.1997 0 TD
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F7 1 Tf
10.9091 0 0 10.9091 301.594 169.2632 Tm
0.2222 Tc
(|)Tj
/F15 1 Tf
7.9701 0 0 7.9701 321.602 182.9001 Tm
0 Tc
()Tj
/F9 1 Tf
10.9091 0 0 10.9091 317.9577 179.6274 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 318.1151 155.9103 Tm
(k)Tj
/F1 1 Tf
0.5799 0 TD
(=2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 335.5327 169.2632 Tm
(1)Tj
/F8 1 Tf
0.5 0 TD
(/p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 351.9303 173.7673 Tm
(k)Tj
/F1 1 Tf
0.5799 0 TD
(\(1)Tj
/F15 1 Tf
0.9445 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
0.0391 Tc
(/)Tj
/F1 1 Tf
1.2145 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F1 1 Tf
0.5364 0 TD
(\))Tj
ET
118.77 137.889 148.242 -0.398 re
f
BT
/F19 1 Tf
5.9776 0 0 5.9776 131.2359 131.0316 Tm
(1)Tj
/F13 1 Tf
8.9664 0 0 8.9664 135.3873 127.2229 Tm
[(In)-342.6(fact,)-342.5(for)]TJ
/F20 1 Tf
5.1696 0 TD
(z)Tj
/F21 1 Tf
0.8067 0 TD
()Tj
/F13 1 Tf
1.0849 0 TD
(200.)Tj
ET
endstream
endobj
82 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F10 13 0 R
/F13 16 0 R
/F14 22 0 R
/F15 23 0 R
/F16 24 0 R
/F19 83 0 R
/F20 84 0 R
/F21 85 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
87 0 obj
<<
/Length 22109
>>
stream
BT
/F5 1 Tf
7.9701 0 0 7.9701 162.3741 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
[(AN)-377.7(UNCER)87.7(T)87.5(AINTY)-377.5(PRINCIPLE)-377.6(F)29.2(O)0(R)-377.8(ARITHMETIC)-377.4(SEQUENCES)]TJ
/F2 1 Tf
10.9091 0 0 10.9091 478.5006 726.9741 Tm
(21)Tj
-32.9726 -2.3745 TD
[(and)-270(the)-270.1(claim)-270(follo)27.8(ws.)-423.3(F)83.4(o)0(r)-270.1(small)]TJ
/F8 1 Tf
14.0793 0 TD
0.2778 Tc
(p<)Tj
/F2 1 Tf
1.8364 0 TD
0 Tc
(101)Tj
/F1 1 Tf
7.9701 0 0 7.9701 308.7891 705.0302 Tm
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 313.5213 701.071 Tm
[(,)-282.7(set)]TJ
/F8 1 Tf
2.0584 0 TD
(K)Tj
/F2 1 Tf
1.1986 0 TD
[(=)-277.7([)0.1(log)]TJ
/F8 1 Tf
2.7916 0 TD
0.044 Tc
(z/)Tj
/F2 1 Tf
1.009 0 TD
0 Tc
[(\(2)-166.6(log)]TJ
/F8 1 Tf
2.5139 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
[(\)])-270.1(and)-270(observ)27.8(e)]TJ
-27.9241 -1.2694 TD
[(that)-284.9(for)]TJ
/F8 1 Tf
3.6004 0 TD
(k)Tj
/F7 1 Tf
0.8299 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(K)Tj
/F2 1 Tf
1.2058 0 TD
[(the)-284.9(n)27.7(u)0(m)27.8(b)-27.7(ers)]TJ
/F8 1 Tf
5.6616 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 258.9001 685.587 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 263.465 687.2229 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 273.1965 691.1821 Tm
(k)Tj
/F2 1 Tf
10.9091 0 0 10.9091 278.3163 687.2229 Tm
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 293.5023 691.1821 Tm
(k)Tj
/F1 1 Tf
0.5799 0 TD
(\(1+)Tj
/F14 1 Tf
1.7707 0 TD
(s/)Tj
/F1 1 Tf
1.1997 0 TD
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F1 1 Tf
0.5364 0 TD
(\))Tj
/F2 1 Tf
10.9091 0 0 10.9091 345.3241 687.2229 Tm
[(all)-284.9(ha)27.8(v)27.8(e)-284.9(argumen)27.9(t)-285(i)0(n)-284.8(the)-284.9(range)]TJ
-20.7648 -1.2694 TD
([0)Tj
/F8 1 Tf
0.7778 0 TD
0.1666 Tc
[(,)130.7(/)]TJ
/F2 1 Tf
1.5503 0 TD
0 Tc
[(2].)-444.4(Hence)-333.3(the)-333.3(left)-333.3(side)-333.3(of)-333.2(\(3.3\))-333.3(exceeds,)-333.2(when)]TJ
/F8 1 Tf
19.9556 0 TD
(q)Tj
/F2 1 Tf
0.76 0 TD
0.2778 Tc
(=1)Tj
/F8 1 Tf
1.5556 0 TD
0 Tc
(/p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 398.0991 677.334 Tm
(1)Tj
/F15 1 Tf
0.5312 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
0.0391 Tc
(/)Tj
/F1 1 Tf
1.2144 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 435.722 673.3749 Tm
(,)Tj
/F9 1 Tf
-26.2609 -1.1503 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F2 1 Tf
0.3333 0.0501 TD
0.2222 Tc
(1+)Tj
/F8 1 Tf
2.7079 0.682 TD
0 Tc
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 187.7579 654.0852 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 192.3228 655.7211 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
172.89 651.166 42.995 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 172.8588 640.3358 Tm
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 178.3472 643.5029 Tm
(1+)Tj
/F14 1 Tf
1.3577 0 TD
(s/)Tj
/F1 1 Tf
1.1996 0 TD
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F9 1 Tf
10.9091 0 0 10.9091 217.0491 660.8259 Tm
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F7 1 Tf
0.5555 0.0501 TD
()Tj
/F9 1 Tf
1.1447 0.95 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 234.0184 634.927 Tm
0.0266 Tc
[(k>)26.6(K)]TJ
/F2 1 Tf
10.9091 0 0 10.9091 258.408 655.6603 Tm
0 Tc
(1)Tj
ET
255.93 651.166 10.381 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 255.9452 640.7941 Tm
(q)Tj
/F14 1 Tf
7.9701 0 0 7.9701 261.2065 643.9453 Tm
(k)Tj
/F7 1 Tf
10.9091 0 0 10.9091 270.5515 648.2809 Tm
()Tj
/F9 1 Tf
1.0556 1.15 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F2 1 Tf
0.3333 0.0501 TD
0.2222 Tc
(1+)Tj
/F8 1 Tf
2.7079 0.682 TD
0 Tc
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 320.5848 654.0852 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 325.1497 655.7211 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
305.73 651.166 42.995 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 305.6867 640.3358 Tm
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 311.1751 643.5029 Tm
(1+)Tj
/F14 1 Tf
1.3576 0 TD
(s/)Tj
/F1 1 Tf
1.1997 0 TD
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F9 1 Tf
10.9091 0 0 10.9091 349.876 660.8259 Tm
()Tj
0 -0.6 TD
()Tj
T*
()Tj
0.5 1.46 TD
()Tj
/F2 1 Tf
0.7362 -1.41 TD
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F2 1 Tf
3.6807 0.6764 TD
(1)Tj
ET
1 i
383.37 650.73 61.551 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 383.3445 640.797 Tm
(q)Tj
/F14 1 Tf
7.9701 0 0 7.9701 388.6057 643.9492 Tm
(K)Tj
/F15 1 Tf
0.9668 0 TD
()Tj
/F1 1 Tf
0.8264 0 TD
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 407.6294 640.797 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F7 1 Tf
0.7046 0 TD
()Tj
/F2 1 Tf
1 0 TD
(1\))Tj
/F1 1 Tf
7.9701 0 0 7.9701 440.1634 643.9492 Tm
(2)Tj
/F9 1 Tf
10.9091 0 0 10.9091 446.0912 663.6623 Tm
()Tj
/F8 1 Tf
0.9027 -1.41 TD
(,)Tj
/F2 1 Tf
-30.9045 -2.8042 TD
[(whic)27.8(h)-333.3(implies)-333.2(\(3.3\))-333.3(for)]TJ
/F8 1 Tf
10.175 0 TD
(z)Tj
/F7 1 Tf
0.7868 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(101)Tj
/F1 1 Tf
7.9701 0 0 7.9701 266.2615 621.6478 Tm
(6)Tj
/F2 1 Tf
10.9091 0 0 10.9091 270.9938 617.6897 Tm
(.)Tj
-11.942 -1.2694 TD
[(Observ)27.8(e)-333.3(that)-333.3(if)]TJ
/F7 1 Tf
6.925 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(w)Tj
/F7 1 Tf
0.7428 0 TD
0.2777 Tc
(|)Tj
/F2 1 Tf
1.6111 0 TD
0 Tc
(2)Tj
/F15 1 Tf
7.9701 0 0 7.9701 250.4269 607.7997 Tm
()Tj
/F1 1 Tf
0.8264 0 TD
(1)Tj
/F14 1 Tf
0.5312 0 TD
(/)Tj
/F1 1 Tf
0.5312 0 TD
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 273.8501 603.8416 Tm
(then)Tj
-10.0086 -2.5149 TD
(log)Tj
/F7 1 Tf
1.4584 0 TD
(|)Tj
/F2 1 Tf
0.2778 0 TD
0.2222 Tc
(1+)Tj
/F8 1 Tf
1.7222 0 TD
0 Tc
(w)Tj
/F7 1 Tf
0.7428 0 TD
(|)Tj
/F2 1 Tf
0.5555 0 TD
0.2777 Tc
[(=R)277.7(e)-55.6(\()]TJ
/F8 1 Tf
2.9584 0 TD
0 Tc
(w)Tj
/F7 1 Tf
0.965 0 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 274.0115 590.0424 Tm
()Tj
/F9 1 Tf
10.9091 0 0 10.9091 270.3671 586.7697 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 270.2665 563.4521 Tm
(n)Tj
/F1 1 Tf
0.6447 0 TD
0.0001 Tc
(=2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 286.2257 576.4065 Tm
0 Tc
(\()Tj
/F7 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.7778 0 TD
(1\))Tj
/F14 1 Tf
7.9701 0 0 7.9701 308.6496 580.9106 Tm
(n)Tj
/F8 1 Tf
10.9091 0 0 10.9091 314.2864 576.4065 Tm
(w)Tj
/F14 1 Tf
7.9701 0 0 7.9701 322.39 580.9106 Tm
(n)Tj
/F8 1 Tf
10.9091 0 0 10.9091 328.0259 576.4065 Tm
(/n)Tj
/F2 1 Tf
1.1003 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
[(Re)-333.3(\()]TJ
/F8 1 Tf
1.9027 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F7 1 Tf
0.5 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(w)Tj
/F7 1 Tf
0.7428 0 TD
(|)Tj
/F1 1 Tf
7.9701 0 0 7.9701 424.8718 580.9106 Tm
(2)Tj
/F8 1 Tf
10.9091 0 0 10.9091 429.604 576.4065 Tm
(/)Tj
/F2 1 Tf
0.5 0 TD
(4)Tj
/F8 1 Tf
0.5 0 TD
(.)Tj
/F2 1 Tf
-29.4904 -2.6378 TD
[(F)83.4(rom)-333.3(this)-333.3(observ)55.6(ation)-333.3(and)-333.3(our)-333.3(claim)-333.3(\(3.3\))-333.3(w)27.8(e)-333.4(deduce)-333.3(easily)-333.2(that)]TJ
0.7754 -2.3004 TD
(log)Tj
/F9 1 Tf
1.4584 1.15 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F2 1 Tf
0.3333 0.05 TD
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F2 1 Tf
2.8301 0.6765 TD
(1)Tj
ET
166.77 525.526 42.995 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 166.7875 514.5913 Tm
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 172.276 517.7584 Tm
(1+)Tj
/F14 1 Tf
1.3578 0 TD
(s/)Tj
/F1 1 Tf
1.1996 0 TD
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F9 1 Tf
10.9091 0 0 10.9091 210.9778 535.0814 Tm
()Tj
0 -0.6 TD
()Tj
T*
()Tj
0.3333 1.2 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F2 1 Tf
0.3333 0.05 TD
0.2222 Tc
(1+)Tj
/F8 1 Tf
2.7079 0.6821 TD
0 Tc
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 253.1318 528.3397 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 257.6967 529.9766 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
238.29 525.526 42.995 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 238.2336 514.5913 Tm
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 243.722 517.7584 Tm
(1+)Tj
/F14 1 Tf
1.3576 0 TD
(s/)Tj
/F1 1 Tf
1.1998 0 TD
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 284.8478 522.5355 Tm
(+)Tj
/F8 1 Tf
1 0 TD
0.1667 Tc
(...)Tj
/F9 1 Tf
1.3333 1.15 TD
0 Tc
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F7 1 Tf
0.6111 0.05 TD
()Tj
/F2 1 Tf
1.0556 0 TD
(Re)Tj
/F9 1 Tf
1.6805 1.41 TD
()Tj
/F8 1 Tf
0.8606 -0.7279 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 361.5472 528.3397 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 366.1121 529.9766 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
0.9999 0 TD
(1)Tj
ET
356.01 525.526 42.995 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 356.0428 514.5913 Tm
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 361.5312 517.7584 Tm
(1+)Tj
/F14 1 Tf
1.3577 0 TD
(s/)Tj
/F1 1 Tf
1.1996 0 TD
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F9 1 Tf
10.9091 0 0 10.9091 400.2332 537.9178 Tm
()Tj
/F7 1 Tf
0.9583 -1.41 TD
()Tj
/F2 1 Tf
3.3315 0.6765 TD
(5)Tj
ET
422.85 525.526 53.933 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 422.7915 514.5913 Tm
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 428.281 517.7584 Tm
(2\(1)Tj
/F15 1 Tf
1.4756 0 TD
()Tj
/F14 1 Tf
0.8265 0 TD
0.039 Tc
(/)Tj
/F1 1 Tf
1.2144 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F1 1 Tf
0.5365 0 TD
(\))Tj
/F8 1 Tf
10.9091 0 0 10.9091 477.9198 522.5355 Tm
(.)Tj
/F2 1 Tf
-32.9194 -2.2448 TD
[(It)-333.4(follo)27.8(ws)-333.3(that)]TJ
0.8091 -1.7784 TD
(log)Tj
/F7 1 Tf
1.4584 0 TD
(|)Tj
/F8 1 Tf
0.2777 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 155.1432 477.0102 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 159.7081 478.6471 Tm
[(\(1)-222.1(+)]TJ
/F8 1 Tf
2.111 0 TD
(s/)Tj
/F2 1 Tf
1.1354 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
-5.4477 -2.1074 TD
0.2777 Tc
()Tj
/F2 1 Tf
1.8333 0 TD
0 Tc
(Re)Tj
/F9 1 Tf
1.6805 0.9501 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 202.2705 441.861 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
(q)Tj
/F9 1 Tf
10.9091 0 0 10.9091 217.3091 471.0396 Tm
()Tj
/F2 1 Tf
0.8457 -0.7279 TD
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F8 1 Tf
1 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 250.663 461.4615 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 255.2279 463.0984 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
226.53 458.566 42.667 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 245.1238 448.1743 Tm
(p)Tj
/F9 1 Tf
2.3167 2.096 TD
()Tj
/F8 1 Tf
0.9028 -1.41 TD
(p)Tj
/F15 1 Tf
7.9701 0 0 7.9701 285.7345 460.1614 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
(s/)Tj
/F1 1 Tf
1.1996 0 TD
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F7 1 Tf
10.9091 0 0 10.9091 321.4296 455.6573 Tm
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(J)Tj
/F2 1 Tf
0.6508 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
-17.6925 -3.3626 TD
(=)Tj
/F8 1 Tf
1.0555 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
0.2223 Tc
(\)+)Tj
/F9 1 Tf
1.6111 0.95 TD
0 Tc
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 212.5001 405.1786 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
(q)Tj
/F9 1 Tf
10.9091 0 0 10.9091 227.5377 434.3562 Tm
()Tj
/F2 1 Tf
0.8458 -0.728 TD
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F8 1 Tf
1 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 260.8926 424.7791 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 265.4575 426.4149 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
236.73 421.966 42.667 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 255.3534 411.4919 Tm
(p)Tj
/F9 1 Tf
2.3167 2.0959 TD
()Tj
/F8 1 Tf
0.9027 -1.41 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 295.9631 423.4789 Tm
0.0391 Tc
(/)Tj
/F1 1 Tf
1.2145 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F9 1 Tf
10.9091 0 0 10.9091 322.7666 431.0834 Tm
( )Tj
/F7 1 Tf
0.8194 -1.11 TD
()Tj
/F2 1 Tf
1 0 TD
(1)Tj
/F7 1 Tf
0.7223 0 TD
()Tj
/F2 1 Tf
1 0 TD
(cos)Tj
/F9 1 Tf
1.5055 1.11 TD
( )Tj
/F8 1 Tf
0.7068 -0.4334 TD
()Tj
/F2 1 Tf
0.7726 0 TD
(log)Tj
/F8 1 Tf
1.4583 0 TD
(p)Tj
ET
385.53 421.966 29.826 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 389.7196 411.4919 Tm
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F9 1 Tf
1.002 1.7959 TD
-0.0001 Tc
(\012\012)Tj
/F7 1 Tf
1.4166 -1.11 TD
0 Tc
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(.)Tj
/F2 1 Tf
-32.1232 -2.9813 TD
(Since)Tj
/F7 1 Tf
2.5 0 TD
()Tj
/F2 1 Tf
0.7778 0 TD
(1)Tj
/F7 1 Tf
0.5 0 TD
()Tj
/F2 1 Tf
0.7777 0 TD
(cos\()Tj
/F8 1 Tf
1.7279 0 TD
()Tj
/F2 1 Tf
0.7725 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(p/)Tj
/F2 1 Tf
1.1698 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
0.2777 Tc
()Tj
/F2 1 Tf
1.8333 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 284.0309 390.4099 Tm
(2)Tj
/F8 1 Tf
10.9091 0 0 10.9091 288.7632 386.4518 Tm
(/)Tj
/F2 1 Tf
0.5 0 TD
[(2\)\(log)-13.8(\()]TJ
/F8 1 Tf
2.9584 0 TD
0.0439 Tc
[(z/)43.9(p)]TJ
/F2 1 Tf
1.5121 0 TD
0 Tc
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/)Tj
/F2 1 Tf
0.6667 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\))Tj
/F1 1 Tf
7.9701 0 0 7.9701 380.2062 390.4099 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 384.9385 386.4518 Tm
[(,)-244.4(w)27.7(e)-222.2(deduce)-222.1(the)-222.2(prop)-27.7(o-)]TJ
-24.3961 -1.2694 TD
(sition.)Tj
ET
482.61 379.329 6.775 -0.398 re
f
482.61 379.028 0.399 -5.978 re
f
488.97 379.028 0.399 -5.978 re
f
482.61 372.969 6.775 -0.398 re
f
BT
10.9091 0 0 10.9091 140.7173 349.7066 Tm
[(W)83.3(e)-333.3(are)-333.4(no)27.8(w)-333.3(ready)-333.3(to)-333.4(pro)27.8(v)27.8(e)-333.3(Theorem)-333.3(3.1.)]TJ
/F16 1 Tf
0 -2.1826 TD
[(Pr)51.2(o)51.1(o)0(f)-370.2(o)0(f)-370.2(The)51.2(or)51.1(em)]TJ
/F2 1 Tf
8.05 0 TD
[(3.1.)-969.8(The)-346.8(rst)-346.8(part)-346.8(of)-346.8(the)-346.8(result)-346.8(w)27.8(a)0(s)-346.8(pro)27.8(v)27.8(e)0(d)-346.8(i)0(n)-346.8(Prop)-27.7(o-)]TJ
-10.0591 -1.2694 TD
[(sition)-421.9(3.6.)-710.2(No)27.7(w,)-444.1(let)]TJ
/F8 1 Tf
9.2817 0 TD
(I)Tj
/F1 1 Tf
7.9701 0 0 7.9701 225.7066 316.007 Tm
(+)Tj
/F2 1 Tf
10.9091 0 0 10.9091 237.3948 312.0478 Tm
(\(and)Tj
/F8 1 Tf
2.4219 0 TD
(I)Tj
/F15 1 Tf
7.9701 0 0 7.9701 269.4675 316.007 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 276.5519 312.0478 Tm
[(\))-422(denote)-421.9(the)-421.9(set)-422(of)-421.9(v)55.5(alues)]TJ
/F8 1 Tf
11.8763 0 TD
(u)Tj
/F2 1 Tf
0.9944 0 TD
(with)Tj
/F8 1 Tf
2.3664 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
(\))Tj
/F7 1 Tf
0.8143 0 TD
()Tj
/F2 1 Tf
1.2033 0 TD
(0)Tj
-33.4726 -1.2694 TD
[(\(resp)-27.8(ectiv)27.9(ely)]TJ
/F8 1 Tf
5.8089 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\))Tj
/F8 1 Tf
0.6667 0 TD
(<)Tj
/F2 1 Tf
1.0556 0 TD
[(0\).)-433.5(T)83.4(aking)]TJ
/F8 1 Tf
4.9006 0 TD
(s)Tj
/F2 1 Tf
0.7466 0 TD
(=)Tj
/F7 1 Tf
1.0555 0 TD
()Tj
/F8 1 Tf
0.7778 0 TD
()Tj
/F2 1 Tf
0.784 0 TD
[(in)-300.5(\(3.2\))-300.5(w)27.8(e)-300.6(deduce)-300.5(that)-300.5(for)]TJ
/F8 1 Tf
11.8616 0 TD
()Tj
/F7 1 Tf
0.7613 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 460.6994 302.4956 Tm
(2)Tj
ET
460.65 301.126 4.234 -0.436 re
f
BT
7.9701 0 0 7.9701 460.6994 294.4379 Tm
(3)Tj
/F2 1 Tf
10.9091 0 0 10.9091 467.9472 298.1997 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
-25.901 -1.2216 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
0.5 1.4111 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 217.6638 284.0209 Tm
()Tj
/F1 1 Tf
-0.6082 -2.7084 TD
(0)Tj
/F8 1 Tf
10.9091 0 0 10.9091 228.4483 272.3278 Tm
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 233.5283 276.8319 Tm
0.0391 Tc
(u)Tj
/F8 1 Tf
10.9091 0 0 10.9091 242.9629 272.3278 Tm
0 Tc
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(du)Tj
/F9 1 Tf
1.093 1.1499 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F7 1 Tf
0.6111 0.0501 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(2)Tj
/F9 1 Tf
0.5 1.1499 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F8 1 Tf
0.4429 0.7265 TD
()Tj
/F2 1 Tf
0.5113 0 TD
(\(1)Tj
/F7 1 Tf
1.111 0 TD
()Tj
/F8 1 Tf
1 0 TD
0.0461 Tc
(/)Tj
/F2 1 Tf
1.1502 0 TD
0 Tc
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
ET
306.81 275.326 69.89 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 321.6757 264.8448 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F7 1 Tf
0.7312 0 TD
()Tj
/F8 1 Tf
1 0 TD
()Tj
/F9 1 Tf
1.9598 1.8359 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F7 1 Tf
0.6111 0.0501 TD
()Tj
/F2 1 Tf
1.1651 0.6764 TD
(6)Tj
ET
397.17 275.326 5.455 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 397.3181 264.8448 Tm
()Tj
0.6014 0.6859 TD
(,)Tj
/F2 1 Tf
-26.1322 -2.2571 TD
[(since)-412.5(0)]TJ
/F7 1 Tf
3.4391 0 TD
()Tj
/F8 1 Tf
1.1877 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 177.8501 246.0682 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 182.4149 247.7051 Tm
(\(1)Tj
/F7 1 Tf
1.1639 0 TD
()Tj
/F8 1 Tf
1.0529 0 TD
0.046 Tc
(/)Tj
/F2 1 Tf
1.1501 0 TD
0 Tc
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
0.1667 Tc
(,G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 258.2755 246.0682 Tm
0 Tc
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 262.8404 247.7051 Tm
(\(1\))Tj
/F7 1 Tf
1.6876 0 TD
()Tj
/F2 1 Tf
1.1877 0 TD
[(1,)-432.4(and)-412.6(since)-412.5(\(b)27.8(y)]TJ
/F8 1 Tf
7.6201 0 TD
()Tj
/F2 1 Tf
0.5113 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/w)Tj
/F2 1 Tf
1.6527 0 TD
0.4099 Tc
(=1)Tj
/F8 1 Tf
1.6877 0 TD
0 Tc
(/)Tj
/F2 1 Tf
0.5 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F7 1 Tf
1.0179 0 TD
()Tj
/F2 1 Tf
1.0528 0 TD
(1\))Tj
/F7 1 Tf
1.1639 0 TD
()Tj
/F9 1 Tf
-33.1948 -0.4639 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 126.0722 239.4899 Tm
()Tj
/F1 1 Tf
-0.2661 -1.1875 TD
(1)Tj
/F7 1 Tf
10.9091 0 0 10.9091 135.0386 233.8571 Tm
({)Tj
/F8 1 Tf
0.5 0 TD
(x)Tj
/F7 1 Tf
0.5715 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(x)Tj
/F15 1 Tf
7.9701 0 0 7.9701 158.4179 237.8152 Tm
()Tj
/F1 1 Tf
0.8264 0 TD
(1)Tj
/F15 1 Tf
0.5312 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(w)Tj
/F8 1 Tf
10.9091 0 0 10.9091 182.6142 233.8571 Tm
(dw)Tj
/F2 1 Tf
1.2633 0 TD
(\))Tj
/F9 1 Tf
-6.6958 -1.228 TD
()Tj
0 -0.6001 TD
()Tj
T*
()Tj
/F8 1 Tf
0.4429 0.7266 TD
()Tj
/F2 1 Tf
0.5112 0 TD
(\(1)Tj
/F7 1 Tf
1.1111 0 TD
()Tj
/F8 1 Tf
1 0 TD
0.0461 Tc
(/)Tj
/F2 1 Tf
1.1501 0 TD
0 Tc
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\))Tj
ET
128.13 210.886 66.859 -0.436 re
f
BT
10.9091 0 0 10.9091 137.335 200.4313 Tm
(\(log)Tj
/F8 1 Tf
1.8472 0 TD
(z)Tj
/F7 1 Tf
0.7312 0 TD
()Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F9 1 Tf
1.3374 1.836 TD
()Tj
0 -0.6001 TD
()Tj
T*
()Tj
/F2 1 Tf
0.6111 0.0501 TD
(=)Tj
/F9 1 Tf
1.0555 1.15 TD
()Tj
0 -0.6001 TD
()Tj
T*
()Tj
/F7 1 Tf
0.5556 0.0501 TD
()Tj
/F2 1 Tf
1.1096 0.6765 TD
(1)Tj
ET
232.53 210.886 5.455 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 232.6745 200.4313 Tm
()Tj
/F7 1 Tf
0.8236 0.6859 TD
()Tj
/F2 1 Tf
1.8433 0.6765 TD
(1)Tj
ET
253.77 210.886 21.463 -0.436 re
f
BT
10.9091 0 0 10.9091 253.7634 200.4313 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.7852 2.047 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 289.1477 219.6084 Tm
()Tj
/F1 1 Tf
-0.6083 -2.7084 TD
(1)Tj
/F7 1 Tf
10.9091 0 0 10.9091 298.1151 207.9143 Tm
({)Tj
/F8 1 Tf
0.5 0 TD
(x)Tj
/F7 1 Tf
0.5715 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(x)Tj
/F15 1 Tf
7.9701 0 0 7.9701 321.4934 212.4184 Tm
()Tj
/F1 1 Tf
0.8264 0 TD
(2+)Tj
/F14 1 Tf
1.3576 0 TD
0.0391 Tc
(/)Tj
/F1 1 Tf
1.2145 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F8 1 Tf
10.9091 0 0 10.9091 365.7036 207.9143 Tm
(dx)Tj
/F9 1 Tf
1.092 1.15 TD
()Tj
0 -0.6001 TD
()Tj
T*
()Tj
/F7 1 Tf
0.6111 0.0501 TD
()Tj
/F2 1 Tf
1.1651 0.6765 TD
(1)Tj
ET
397.05 210.886 5.455 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 397.0829 200.4313 Tm
()Tj
/F2 1 Tf
0.8236 0.6859 TD
(+)Tj
2.6962 0.6765 TD
(1)Tj
ET
418.17 210.886 40.07 -0.436 re
f
BT
10.9091 0 0 10.9091 418.1719 200.4313 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F7 1 Tf
0.7312 0 TD
()Tj
/F8 1 Tf
1 0 TD
()Tj
/F7 1 Tf
0.8709 0.6859 TD
()Tj
/F2 1 Tf
1.1652 0.6765 TD
(3)Tj
ET
475.17 210.886 5.455 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 475.2688 200.4313 Tm
()Tj
0.6014 0.6859 TD
(.)Tj
/F2 1 Tf
-33.2777 -2.2307 TD
[(Com)27.8(bining)-333.3(this)-333.2(with)-333.3(Prop)-27.7(osition)-333.2(3.7)-333.3(w)27.8(e)-333.4(deduce)-333.3(easily)-333.2(that)]TJ
/F9 1 Tf
2.8747 -0.8046 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 156.2202 150.0613 Tm
(I)Tj
/F11 1 Tf
4.9813 0 0 4.9813 160.5141 152.3307 Tm
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 168.7253 159.9532 Tm
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 173.8042 164.4573 Tm
0.0391 Tc
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 183.2389 159.9532 Tm
0 Tc
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(du)Tj
/F7 1 Tf
1.2596 0 TD
()Tj
/F8 1 Tf
3.0408 0.6765 TD
()Tj
ET
237.93 162.886 48.622 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 237.9517 152.4702 Tm
(2\()Tj
/F8 1 Tf
0.8889 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 252.9235 155.6214 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 260.0797 152.4702 Tm
(+)Tj
/F8 1 Tf
1 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 277.599 155.6214 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 282.3313 152.4702 Tm
(\))Tj
/F9 1 Tf
0.4984 1.796 TD
( )Tj
/F2 1 Tf
0.7639 -1.1101 TD
(exp)Tj
/F7 1 Tf
1.5278 0 TD
({)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 369.7943 158.3173 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 374.5266 159.9532 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
0.2223 Tc
(})Tj
/F2 1 Tf
1.7222 0 TD
0 Tc
(1)Tj
/F9 1 Tf
0.5 1.1101 TD
(\012)Tj
/F7 1 Tf
0.8195 -1.1101 TD
()Tj
/F2 1 Tf
1.1096 0.6765 TD
(3)Tj
ET
473.25 162.886 5.455 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 473.3081 152.4702 Tm
()Tj
/F2 1 Tf
-32.4966 0.6859 TD
(\(3.4\))Tj
/F7 1 Tf
9.8682 -2.6222 TD
()Tj
/F2 1 Tf
1.2958 0.6765 TD
(1)Tj
ET
237.93 134.326 10.729 -0.436 re
f
BT
10.9091 0 0 10.9091 237.9517 123.8645 Tm
(5)Tj
/F8 1 Tf
0.5 0 TD
()Tj
/F2 1 Tf
0.7598 0.6859 TD
(exp)Tj
/F7 1 Tf
1.5278 0 TD
({)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 325.3858 129.7116 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 330.1181 131.3474 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(.)Tj
ET
endstream
endobj
88 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F11 14 0 R
/F14 22 0 R
/F15 23 0 R
/F16 24 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
91 0 obj
<<
/Length 21007
>>
stream
BT
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
(22)Tj
/F5 1 Tf
7.9701 0 0 7.9701 202.119 726.9739 Tm
[(ANDREW)-377.5(GRANVILLE)-377.4(AND)-377.7(K)0(.)-377.7(SOUND)29.4(ARARAJAN)]TJ
/F2 1 Tf
10.9091 0 0 10.9091 140.7173 701.071 Tm
(Put)Tj
/F8 1 Tf
1.9584 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 168.3258 699.4351 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 176.0887 701.071 Tm
(=)Tj
/F8 1 Tf
1.0555 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
0.2222 Tc
[(\))222.2(\()222.2(1+2)]TJ
/F8 1 Tf
3 0 TD
0 Tc
()Tj
/F2 1 Tf
0.5503 0 TD
[(\).)-444.5(Then)-333.2(b)27.7(y)-333.3(Prop)-27.7(osition)-333.2(3.6)-333.3(w)27.8(e)-333.4(get)-333.3(that)]TJ
/F9 1 Tf
-6.6801 -0.9312 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 183.8416 687.7589 Tm
()Tj
/F14 1 Tf
-0.6082 -2.7084 TD
(u)Tj
/F12 1 Tf
4.9813 0 0 4.9813 183.8964 665.1766 Tm
(1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 194.6271 676.0648 Tm
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 199.7061 680.5689 Tm
0.0391 Tc
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 209.1407 676.0648 Tm
0 Tc
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(du)Tj
/F7 1 Tf
1.2596 0 TD
()Tj
/F8 1 Tf
0.9445 0 TD
(e)Tj
/F15 1 Tf
7.9701 0 0 7.9701 267.738 680.5689 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
0.1155 Tc
(u)Tj
/F12 1 Tf
4.9813 0 0 4.9813 283.9053 679.5726 Tm
0 Tc
(1)Tj
/F9 1 Tf
10.9091 0 0 10.9091 290.1101 690.9131 Tm
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 301.0192 687.7589 Tm
()Tj
/F1 1 Tf
-0.6084 -2.7084 TD
(0)Tj
/F8 1 Tf
10.9091 0 0 10.9091 311.8037 676.0648 Tm
(e)Tj
/F1 1 Tf
7.9701 0 0 7.9701 316.8837 680.5689 Tm
(\()Tj
/F14 1 Tf
0.4131 0 TD
()Tj
/F1 1 Tf
0.5061 0 TD
(+)Tj
/F14 1 Tf
0.8264 0 TD
()Tj
/F1 1 Tf
0.587 0 TD
(\))Tj
/F14 1 Tf
0.4132 0 TD
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 344.1694 676.0648 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(du)Tj
/F7 1 Tf
-11.118 -2.6844 TD
()Tj
/F2 1 Tf
1.0541 0.6765 TD
(3)Tj
ET
263.85 649.726 5.455 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 263.9432 639.2977 Tm
()Tj
/F2 1 Tf
0.768 0.6859 TD
(exp)Tj
/F7 1 Tf
1.5278 0 TD
({)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.7057 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
0.5504 0 TD
(\))Tj
/F7 1 Tf
0.611 0 TD
()Tj
/F8 1 Tf
1 0 TD
0.1132 Tc
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 363.0765 645.1447 Tm
0 Tc
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 370.2326 646.7806 Tm
0.2222 Tc
(+5)Tj
/F8 1 Tf
1.5 0 TD
0 Tc
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.7057 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
0.5504 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(.)Tj
/F2 1 Tf
-28.7326 -2.3699 TD
[(No)27.8(w)]TJ
/F8 1 Tf
2.2205 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.5911 0 TD
(+)Tj
/F8 1 Tf
0.8855 0 TD
()Tj
/F2 1 Tf
0.5503 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 203.199 624.8867 Tm
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 208.3756 620.9276 Tm
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F2 1 Tf
1.0556 0 TD
[(\(1)-107.6(+)]TJ
/F8 1 Tf
1.8819 0 TD
()Tj
/F2 1 Tf
0.658 0 TD
(+)Tj
/F8 1 Tf
0.8854 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 290.0055 624.8867 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 294.7377 620.9276 Tm
(\))Tj
/F8 1 Tf
0.389 0 TD
(H)Tj
/F2 1 Tf
0.9124 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
[(\),)-287.4(and)]TJ
/F8 1 Tf
2.8414 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.5911 0 TD
(+)Tj
/F8 1 Tf
0.8854 0 TD
()Tj
/F2 1 Tf
0.5504 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(e)Tj
/F1 1 Tf
7.9701 0 0 7.9701 406.7676 624.8867 Tm
(2)Tj
/F14 1 Tf
0.5314 0 TD
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 416.1783 620.9276 Tm
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(2)Tj
/F8 1 Tf
0.5001 0 TD
(.)Tj
/F2 1 Tf
0.2777 0 TD
(8)Tj
/F8 1 Tf
0.5 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\).)Tj
-33.3059 -1.2694 TD
(Hence)Tj
/F8 1 Tf
3.0216 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.7386 0 TD
(+)Tj
/F8 1 Tf
1.0329 0 TD
()Tj
/F2 1 Tf
0.5503 0 TD
0.2551 Tc
(\)+5)Tj
/F8 1 Tf
2.1768 0 TD
0 Tc
(J)Tj
/F2 1 Tf
0.6508 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.7386 0 TD
(+)Tj
/F8 1 Tf
1.0329 0 TD
()Tj
/F2 1 Tf
0.5503 0 TD
(\))Tj
/F7 1 Tf
0.6439 0 TD
()Tj
/F8 1 Tf
1.0329 0 TD
0.1133 Tc
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 282.2466 605.4436 Tm
0 Tc
(1)Tj
/F7 1 Tf
10.9091 0 0 10.9091 290.9061 607.0795 Tm
()Tj
/F8 1 Tf
1.1377 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.644 0 TD
()Tj
/F2 1 Tf
1.0329 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 355.6045 605.4436 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 360.3367 607.0795 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.644 0 TD
()Tj
/F2 1 Tf
1.0328 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(J)Tj
/F2 1 Tf
0.6508 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.644 0 TD
()Tj
/F2 1 Tf
1.0329 0 TD
[(5,)-394.9(and)-382.6(so)-382.6(w)27.8(e)]TJ
-28.3901 -1.2694 TD
[(conclude)-333.3(that)]TJ
/F9 1 Tf
5.3971 -0.6811 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 188.5858 582.6471 Tm
()Tj
/F14 1 Tf
-0.6082 -2.7084 TD
(u)Tj
/F12 1 Tf
4.9813 0 0 4.9813 188.6406 560.0648 Tm
(1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 199.3704 570.9529 Tm
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 204.4503 575.457 Tm
0.0391 Tc
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 213.8849 570.9529 Tm
0 Tc
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(du)Tj
/F7 1 Tf
1.3707 0 TD
()Tj
/F2 1 Tf
1.6569 0.6765 TD
(1)Tj
ET
271.05 573.886 16.183 -0.436 re
f
BT
10.9091 0 0 10.9091 271.0207 563.47 Tm
(20)Tj
/F8 1 Tf
1.0001 0 TD
()Tj
/F2 1 Tf
0.7597 0.6859 TD
(exp)Tj
/F7 1 Tf
1.5278 0 TD
({)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.611 0 TD
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 363.9094 569.3161 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 368.6426 570.9529 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(.)Tj
/F2 1 Tf
-28.2977 0 TD
(\(3.5\))Tj
2.0091 -2.4039 TD
[(Similarly)-333.2(note)-333.3(that,)-333.3(with)]TJ
/F8 1 Tf
11.1973 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 269.1138 543.0924 Tm
(0)Tj
/F2 1 Tf
10.9091 0 0 10.9091 276.8767 544.7292 Tm
(=)Tj
/F8 1 Tf
1.0555 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\)\(1)Tj
/F7 1 Tf
1.5 0 TD
()Tj
/F2 1 Tf
1 0 TD
(2)Tj
/F8 1 Tf
0.5 0 TD
()Tj
/F2 1 Tf
0.5503 0 TD
(\),)Tj
/F9 1 Tf
-16.8118 -0.9311 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 174.1021 531.4162 Tm
(u)Tj
/F12 1 Tf
4.9813 0 0 4.9813 179.0047 530.4199 Tm
(0)Tj
/F1 1 Tf
7.9701 0 0 7.9701 169.2533 509.8301 Tm
(0)Tj
/F8 1 Tf
10.9091 0 0 10.9091 185.2095 519.723 Tm
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 190.2894 524.2272 Tm
0.039 Tc
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 199.724 519.723 Tm
0 Tc
(|)Tj
/F8 1 Tf
0.2777 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
(|)Tj
/F8 1 Tf
0.2777 0 TD
(du)Tj
/F7 1 Tf
1.2597 0 TD
()Tj
/F8 1 Tf
0.9445 0 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 258.3203 524.2272 Tm
0.1155 Tc
(u)Tj
/F12 1 Tf
4.9813 0 0 4.9813 267.9023 523.2309 Tm
0 Tc
(0)Tj
/F9 1 Tf
10.9091 0 0 10.9091 274.1061 534.5714 Tm
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 285.0152 531.4162 Tm
()Tj
/F1 1 Tf
-0.6082 -2.7084 TD
(0)Tj
/F8 1 Tf
10.9091 0 0 10.9091 295.7998 519.723 Tm
(e)Tj
/F1 1 Tf
7.9701 0 0 7.9701 300.8797 524.2272 Tm
(\()Tj
/F14 1 Tf
0.4132 0 TD
()Tj
/F15 1 Tf
0.506 0 TD
()Tj
/F14 1 Tf
0.8265 0 TD
()Tj
/F1 1 Tf
0.5869 0 TD
(\))Tj
/F14 1 Tf
0.4132 0 TD
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 328.1654 519.723 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
(|)Tj
/F8 1 Tf
0.2777 0 TD
(du)Tj
/F7 1 Tf
-10.5142 -2.5931 TD
()Tj
/F2 1 Tf
1.9321 0.6765 TD
(3)Tj
ET
254.49 494.326 24.612 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 254.4369 483.9522 Tm
()Tj
/F7 1 Tf
0.7057 0 TD
()Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
0.8267 0.6859 TD
(exp)Tj
/F7 1 Tf
1.5278 0 TD
({)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F7 1 Tf
0.7058 0 TD
()Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
0.5503 0 TD
0.2223 Tc
(\)+)Tj
/F8 1 Tf
1.6111 0 TD
0.1132 Tc
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 372.8159 489.7993 Tm
0 Tc
(0)Tj
/F2 1 Tf
10.9091 0 0 10.9091 379.9731 491.4351 Tm
0.2222 Tc
(+5)Tj
/F8 1 Tf
1.5 0 TD
0 Tc
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F7 1 Tf
0.7057 0 TD
()Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
0.5503 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(.)Tj
/F2 1 Tf
-29.6255 -2.2861 TD
[(No)27.8(w)]TJ
/F8 1 Tf
2.2778 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F7 1 Tf
0.7058 0 TD
()Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
0.5503 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(J)Tj
/F2 1 Tf
0.6508 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
[(\),)-333.3(and)]TJ
/F8 1 Tf
-2.7515 -2.3378 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F7 1 Tf
0.7057 0 TD
()Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
0.5504 0 TD
0.1666 Tc
(\)=)Tj
/F9 1 Tf
1.5 0.9499 TD
0 Tc
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 242.6989 427.196 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 258.932 448.4324 Tm
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F8 1 Tf
1 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 283.0605 446.7965 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 287.6254 448.4324 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
258.93 443.926 42.667 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 277.5213 433.5093 Tm
(p)Tj
2.3167 0.6859 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 308.283 445.4964 Tm
0.0391 Tc
(/)Tj
/F1 1 Tf
1.2145 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F8 1 Tf
10.9091 0 0 10.9091 335.0864 440.9923 Tm
(p)Tj
/F15 1 Tf
7.9701 0 0 7.9701 340.5749 445.4964 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
0.1155 Tc
(/)Tj
/F1 1 Tf
1.2954 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F7 1 Tf
10.9091 0 0 10.9091 229.857 403.6523 Tm
()Tj
/F9 1 Tf
0.9445 0.95 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 242.6989 389.857 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 258.932 411.0934 Tm
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F8 1 Tf
1 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 283.0605 409.4565 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 287.6254 411.0934 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
258.93 406.606 42.667 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 277.5213 396.1693 Tm
(p)Tj
2.3167 0.6859 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 308.283 408.1564 Tm
0.0391 Tc
(/)Tj
/F1 1 Tf
1.2145 0 TD
0 Tc
(log)Tj
/F14 1 Tf
1.5495 0 TD
(z)Tj
/F9 1 Tf
10.9091 0 0 10.9091 335.0864 415.7619 Tm
( )Tj
/F2 1 Tf
0.5972 -1.11 TD
(1)Tj
/F7 1 Tf
0.7223 0 TD
()Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
0.6628 0.6765 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(p)Tj
ET
367.53 406.606 21.463 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 367.5885 396.1693 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.8408 0.6859 TD
(+)Tj
/F8 1 Tf
1.1096 0.6765 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 410.7786 414.9918 Tm
(2)Tj
ET
404.73 406.606 10.736 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 407.4152 396.1693 Tm
(2)Tj
/F9 1 Tf
0.8517 1.796 TD
(\012)Tj
/F2 1 Tf
-17.1279 -3.9063 TD
(=)Tj
/F8 1 Tf
0.9445 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\)\(1)Tj
/F7 1 Tf
1.5 0 TD
()Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
0.7726 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 312.2461 377.6518 Tm
(2)Tj
/F8 1 Tf
10.9091 0 0 10.9091 316.9783 373.1477 Tm
(/)Tj
/F2 1 Tf
0.5 0 TD
[(2\))-222.2(+)]TJ
/F8 1 Tf
2.1111 0 TD
0.1132 Tc
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 360.535 371.5118 Tm
0 Tc
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 365.2673 373.1477 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
-13.285 -1.5434 TD
()Tj
/F8 1 Tf
0.9445 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\)\(1)Tj
/F7 1 Tf
1.5 0 TD
()Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
0.7726 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 312.2461 360.8149 Tm
(2)Tj
/F8 1 Tf
10.9091 0 0 10.9091 316.9783 356.3109 Tm
(/)Tj
/F2 1 Tf
0.5 0 TD
[(2\))-222.2(+)]TJ
/F8 1 Tf
2.1111 0 TD
()Tj
/F9 1 Tf
0.5503 0.9038 TD
()Tj
ET
1 i
362.37 366.21 51.274 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 362.3761 356.3109 Tm
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 395.1581 354.6749 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 399.8904 356.3109 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(,)Tj
/F2 1 Tf
-27.0279 -1.8338 TD
(since)Tj
/F8 1 Tf
2.4887 0 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 155.0167 334.669 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 159.7489 336.3059 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.7311 0 TD
()Tj
/F9 1 Tf
1.12 0.855 TD
()Tj
ET
200.37 346.126 51.274 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 200.3686 336.3059 Tm
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F8 1 Tf
0.389 0 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 233.1507 334.669 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 237.8829 336.3059 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
[(\))-371.9(b)27.7(y)-371.9(Cauc)27.8(h)27.8(ys)-372(inequalit)27.9(y)83.3(.)-560.4(F)83.4(rom)-372(these)-372(observ)55.6(ations)]TJ
-11.7884 -1.2694 TD
[(and)-373.2(our)-373.2(denition)-373.1(of)]TJ
/F8 1 Tf
9.4679 0 TD
()Tj
/F2 1 Tf
0.9236 0 TD
[(it)-373.2(follo)27.8(ws)-373.2(that)]TJ
/F8 1 Tf
6.5697 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F7 1 Tf
0.7323 0 TD
()Tj
/F8 1 Tf
1.0267 0 TD
()Tj
/F2 1 Tf
0.5503 0 TD
0.2488 Tc
(\)+5)Tj
/F8 1 Tf
2.1643 0 TD
0 Tc
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F7 1 Tf
0.7323 0 TD
()Tj
/F8 1 Tf
1.0267 0 TD
()Tj
/F2 1 Tf
0.5503 0 TD
0.2488 Tc
(\)+)Tj
/F8 1 Tf
1.6643 0 TD
0.1132 Tc
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 433.7684 320.8209 Tm
0 Tc
(0)Tj
/F7 1 Tf
10.9091 0 0 10.9091 442.2556 322.4578 Tm
()Tj
/F8 1 Tf
1.1221 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6377 0 TD
()Tj
/F2 1 Tf
-33.1948 -1.2694 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 133.322 306.9729 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 138.0543 308.6097 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
1 0 TD
[(5,)-333.4(and)-333.3(so)]TJ
/F9 1 Tf
-2.5005 -0.9311 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 188.4244 295.2966 Tm
(u)Tj
/F12 1 Tf
4.9813 0 0 4.9813 193.328 294.3004 Tm
(0)Tj
/F1 1 Tf
7.9701 0 0 7.9701 183.5766 273.7106 Tm
(0)Tj
/F8 1 Tf
10.9091 0 0 10.9091 199.5318 283.6035 Tm
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 204.6117 288.1076 Tm
0.0391 Tc
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 214.0463 283.6035 Tm
0 Tc
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(du)Tj
/F7 1 Tf
1.3707 0 TD
()Tj
/F2 1 Tf
1.6569 0.6764 TD
(1)Tj
ET
271.17 286.606 16.183 -0.436 re
f
BT
10.9091 0 0 10.9091 271.1831 276.1205 Tm
(20)Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
0.7597 0.6859 TD
(exp)Tj
/F7 1 Tf
1.5278 0 TD
({)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 364.0717 281.9666 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 368.804 283.6035 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(.)Tj
/F2 1 Tf
-28.3125 0 TD
(\(3.6\))Tj
2.0091 -2.3126 TD
[(Com)27.8(bining)-333.3(\(3.4\),)-333.3(\(3.5\),)-333.3(and)-333.2(\(3.6\))-333.3(w)27.8(e)-333.4(deduce)-333.3(that)]TJ
/F9 1 Tf
2.1433 -0.9057 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 170.1599 223.7539 Tm
(I)Tj
/F11 1 Tf
4.9813 0 0 4.9813 174.4528 226.0234 Tm
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 180.3477 223.7539 Tm
()Tj
/F1 1 Tf
0.7084 0 TD
([)Tj
/F14 1 Tf
0.2951 0 TD
(u)Tj
/F12 1 Tf
4.9813 0 0 4.9813 193.2483 222.7577 Tm
(0)Tj
/F14 1 Tf
7.9701 0 0 7.9701 197.1367 223.7539 Tm
(,u)Tj
/F12 1 Tf
4.9813 0 0 4.9813 204.3925 222.7577 Tm
(1)Tj
/F1 1 Tf
7.9701 0 0 7.9701 208.2799 223.7539 Tm
(])Tj
/F8 1 Tf
10.9091 0 0 10.9091 212.9494 233.6458 Tm
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 218.0284 238.1499 Tm
0.0391 Tc
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 227.463 233.6458 Tm
0 Tc
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(du)Tj
/F7 1 Tf
1.3707 0 TD
()Tj
/F2 1 Tf
1.6569 0.6765 TD
(1)Tj
ET
284.61 236.566 16.183 -0.436 re
f
BT
10.9091 0 0 10.9091 284.5998 226.1628 Tm
(10)Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
0.7597 0.6859 TD
(exp)Tj
/F7 1 Tf
1.5278 0 TD
({)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.611 0 TD
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F1 1 Tf
7.9701 0 0 7.9701 377.4884 232.0099 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 382.2207 233.6458 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F2 1 Tf
1 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(J)Tj
/F2 1 Tf
0.6507 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(.)Tj
/F2 1 Tf
-29.5424 -2.4952 TD
[(No)27.8(w)]TJ
/F8 1 Tf
2.2778 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 149.8929 204.7901 Tm
(1)Tj
/F7 1 Tf
10.9091 0 0 10.9091 157.0491 206.4259 Tm
()Tj
/F8 1 Tf
1 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 174.2037 204.7901 Tm
(0)Tj
/F7 1 Tf
10.9091 0 0 10.9091 181.9666 206.4259 Tm
()Tj
/F2 1 Tf
1.0555 0 TD
(4)Tj
/F8 1 Tf
0.5 0 TD
0.1132 Tc
(H)Tj
/F2 1 Tf
1.4628 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(2)Tj
/F8 1 Tf
0.5 0 TD
(H)Tj
/F2 1 Tf
0.9125 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4836 0 TD
[(\),)-333.4(so)-333.3(the)-333.3(theorem)-333.3(follo)27.8(ws.)]TJ
ET
482.61 213.249 6.775 -0.399 re
f
482.61 212.828 0.399 -5.978 re
f
488.97 212.828 0.399 -5.978 re
f
482.61 206.769 6.775 -0.398 re
f
BT
10.9091 0 0 10.9091 140.7173 182.6151 Tm
(3b.)Tj
/F16 1 Tf
2.2153 0 TD
[(L)51.1(o)51.1(c)51.1(alization)-491.8(of)-491.9(sign)-492(changes)-491.8(of)]TJ
/F8 1 Tf
14.3569 0 TD
(E)Tj
/F16 1 Tf
0.7958 0 TD
(.)Tj
/F2 1 Tf
1.1886 0 TD
[(W)83.3(e)-479.1(sa)27.8(w)-479.1(i)0(n)-479.2(Corollary)-479(3.3)-479.2(that)]TJ
-20.5657 -1.2694 TD
[(\(in)-321.2(t)27.8(ypical)-321.1(situations\))]TJ
/F8 1 Tf
9.7525 0 TD
(E)Tj
/F2 1 Tf
1.117 0 TD
[(c)27.7(hanges)-321.1(sign)-321.2(in)-321.2(in)27.8(terv)55.6(als)-321.2(of)-321.2(the)-321.2(form)-321.1([)]TJ
/F8 1 Tf
16.354 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\(1)Tj
/F7 1 Tf
1.0868 0 TD
()Tj
/F8 1 Tf
0.9757 0 TD
(A/)Tj
/F2 1 Tf
1.4167 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(,)Tj
-33.6948 -1.2694 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
[(\(1)-37.7(+)]TJ
/F8 1 Tf
1.7419 0 TD
(A/)Tj
/F2 1 Tf
1.4167 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
[(\)].)-413.6(W)83.3(e)-241(consider)-240.9(no)27.8(w)-241.1(the)-241(problem)-240.9(of)-241(pro)27.8(viding)-240.9(a)-241.1(b)-27.7(etter)-241(lo)-27.8(calization)]TJ
-5.7618 -1.2694 TD
[(of)-338.1(the)-337.9(sign)-338(c)27.7(hanges)-338(of)]TJ
/F8 1 Tf
9.7847 0 TD
(E)Tj
/F2 1 Tf
1.1339 0 TD
[(for)-338(small)-338.1(v)55.6(alues)-338(of)]TJ
/F8 1 Tf
8.2827 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
[(.)-458.6(Our)-338(main)-338(result)-338(of)-338(this)-338(section)]TJ
-19.7738 -1.2694 TD
[(is)-333.3(the)-333.3(follo)27.8(wing:)]TJ
ET
endstream
endobj
92 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F11 14 0 R
/F12 15 0 R
/F14 22 0 R
/F15 23 0 R
/F16 24 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
94 0 obj
<<
/Length 19925
>>
stream
BT
/F5 1 Tf
7.9701 0 0 7.9701 162.3741 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
[(AN)-377.7(UNCER)87.7(T)87.5(AINTY)-377.5(PRINCIPLE)-377.6(F)29.2(O)0(R)-377.8(ARITHMETIC)-377.4(SEQUENCES)]TJ
/F2 1 Tf
10.9091 0 0 10.9091 478.5006 726.9741 Tm
(23)Tj
/F5 1 Tf
-30.9635 -2.3745 TD
[(Pr)22.3(oposition)-313.1(3.8)]TJ
/F2 1 Tf
7.7909 0 TD
(.)Tj
/F16 1 Tf
0.669 0 TD
[(With)-301.1(notation)-301(as)-301.2(ab)51.1(ove)]TJ
/F2 1 Tf
9.9523 0 TD
(,)Tj
/F16 1 Tf
0.5902 0 TD
[(supp)51.2(ose)-301.2(that)]TJ
/F2 1 Tf
5.5856 0 TD
(max)Tj
/F14 1 Tf
7.9701 0 0 7.9701 429.2533 699.4351 Tm
(x)Tj
/F15 1 Tf
0.5981 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 447.8267 701.071 Tm
(|)Tj
/F8 1 Tf
0.2777 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
0.1111 Tc
(|)Tj
/F2 1 Tf
-32.5837 -1.2694 TD
0 Tc
(1)Tj
/F8 1 Tf
0.5 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 142.5285 685.587 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 147.0934 687.2229 Tm
[(\(1\))-166.7(log)]TJ
/F8 1 Tf
2.9027 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\)\))Tj
/F16 1 Tf
1.1006 0 TD
[(for)-322.8(some)]TJ
/F8 1 Tf
4.0829 0 TD
(u)Tj
/F7 1 Tf
0.8502 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(6)Tj
/F16 1 Tf
0.5 0 TD
[(.)-448.3(Then)-322.9(ther)51.2(e)-322.8(exist)-322.7(p)51.1(oints)]TJ
/F8 1 Tf
10.9818 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 393.1527 685.587 Tm
(+)Tj
/F8 1 Tf
10.9091 0 0 10.9091 400.2371 687.2229 Tm
0.1667 Tc
(,u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 411.3305 685.587 Tm
0 Tc
()Tj
/F7 1 Tf
10.9091 0 0 10.9091 421.4456 687.2229 Tm
()Tj
/F2 1 Tf
0.9445 0 TD
([)Tj
/F8 1 Tf
0.2777 0 TD
(u)Tj
/F7 1 Tf
0.7187 0 TD
()Tj
/F2 1 Tf
0.9241 0 TD
(1)Tj
/F8 1 Tf
0.5 0 TD
0.1667 Tc
(,u)Tj
/F2 1 Tf
1.1632 0 TD
0.1462 Tc
[(+1)146.2(])]TJ
/F16 1 Tf
-32.2708 -1.2694 TD
0 Tc
[(such)-357.7(that)]TJ
/F8 1 Tf
4.3189 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F1 1 Tf
7.9701 0 0 7.9701 185.0839 671.739 Tm
(+)Tj
/F2 1 Tf
10.9091 0 0 10.9091 192.1684 673.3749 Tm
(\))Tj
/F8 1 Tf
0.3889 0 TD
(,)Tj
/F7 1 Tf
0.4445 0 TD
()Tj
/F8 1 Tf
0.7778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 228.9136 671.739 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 235.998 673.3749 Tm
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(max)Tj
/F14 1 Tf
7.9701 0 0 7.9701 275.0884 671.739 Tm
(x)Tj
/F15 1 Tf
0.5981 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 293.6618 673.3749 Tm
(|)Tj
/F8 1 Tf
0.2777 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
(|)Tj
/F16 1 Tf
0.2777 0 TD
(.)Tj
/F2 1 Tf
-16.7206 -2.1827 TD
[(Prop)-27.7(osition)-261.5(3.8)-261.6(\(whic)27.9(h)-261.5(m)0(a)27.8(y)-261.5(b)-27.8(e)-261.5(easily)-261.5(deduced)-261.5(from)-261.5(the)-261.5(lemmas)-261.4(of)-261.5(this)-261.5(sec-)]TJ
-2.0091 -1.2694 TD
[(tion\))-252.1(can)-252.1(b)-27.8(e)-252.1(used)-252.1(to)-252.1(reduce)-252.1(the)-252.1(size)-252.2(of)-252.1(the)-252.1(in)27.8(terv)55.6(al)-252.1(in)-252.1(Theorem)-252.1(3.1.)-417.4(In)-252.2(Corollary)]TJ
T*
[(3.3)-364(this)-363.9(is)-364(simple)-364(to)-364.1(state:)-505.8(F)83.4(o)0(r)]TJ
/F8 1 Tf
13.7845 0 TD
(e)Tj
/F14 1 Tf
7.9701 0 0 7.9701 274.2556 625.8271 Tm
(A)Tj
/F7 1 Tf
10.9091 0 0 10.9091 284.6844 621.868 Tm
()Tj
/F8 1 Tf
1.1068 0 TD
(u)Tj
/F7 1 Tf
0.9014 0 TD
()Tj
/F2 1 Tf
1.1067 0 TD
[(log)-180.6(log)]TJ
/F8 1 Tf
2.9167 0 TD
0.044 Tc
(z/)Tj
/F2 1 Tf
1.009 0 TD
0 Tc
[(\(2)-166.6(log)-180.6(log)-180.5(log)]TJ
/F8 1 Tf
5.4305 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
[(\))-364(the)-364.1(in)27.8(terv)55.6(al)]TJ
-28.1862 -1.2694 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
(u)Tj
/F7 1 Tf
0.7946 0 TD
()Tj
/F2 1 Tf
1 0 TD
(1)Tj
/F8 1 Tf
0.5 0 TD
0.1667 Tc
(,u)Tj
/F2 1 Tf
1.2392 0 TD
0 Tc
[(+)-222.2(1)0(])-333.3(con)27.8(tains)-333.3(p)-27.7(oin)27.8(ts)]TJ
/F8 1 Tf
9.0389 0 TD
(u)Tj
/F15 1 Tf
7.9701 0 0 7.9701 265.2314 606.3841 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 275.9522 608.02 Tm
[(satisfying)-333.3(the)-333.3(conclusions)-333.2(of)-333.3(Corollary)-333.2(3.3.)]TJ
/F5 1 Tf
-12.3965 -2.1827 TD
[(Lemma)-377.7(3.9)]TJ
/F2 1 Tf
5.1505 0 TD
(.)Tj
/F16 1 Tf
1.1667 0 TD
[(Uniformly)-357.7(for)]TJ
/F8 1 Tf
6.3656 0 TD
0.2778 Tc
(u>)Tj
/F2 1 Tf
1.9058 0 TD
0 Tc
(0,)Tj
/F8 1 Tf
-16.5976 -2.5519 TD
(uE)Tj
/F2 1 Tf
1.3683 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
0.0753 Tc
(\)+)Tj
/F9 1 Tf
1.3174 1.3612 TD
0 Tc
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 169.4944 568.0648 Tm
()Tj
/F14 1 Tf
-0.6084 -2.7084 TD
(u)Tj
/F8 1 Tf
10.9091 0 0 10.9091 180.279 556.3706 Tm
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(t)Tj
/F2 1 Tf
0.3611 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(dt)Tj
/F2 1 Tf
0.957 0 TD
(+)Tj
1.6963 0.6765 TD
(1)Tj
ET
222.33 559.366 21.463 -0.436 re
f
BT
10.9091 0 0 10.9091 222.3273 548.8877 Tm
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F9 1 Tf
0.7853 1.636 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 247.1193 543.2239 Tm
(p)Tj
/F15 1 Tf
0.5348 0 TD
()Tj
/F14 1 Tf
0.8265 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 265.5741 563.8117 Tm
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F8 1 Tf
1 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 289.7026 562.1749 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 294.2675 563.8117 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5032 0 TD
(\))Tj
ET
265.53 559.366 42.667 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 284.1634 548.8877 Tm
(p)Tj
/F2 1 Tf
2.4834 0.6859 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
0.2599 Tc
(pE)Tj
/F9 1 Tf
1.5588 1.11 TD
0 Tc
( )Tj
/F8 1 Tf
0.5973 -1.11 TD
(u)Tj
/F7 1 Tf
0.6478 0 TD
()Tj
/F2 1 Tf
0.9657 0.6765 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(p)Tj
ET
368.25 559.366 21.463 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 368.254 548.8877 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.6185 1.796 TD
(\012)Tj
/F2 1 Tf
0.8751 -1.11 TD
(=)Tj
/F8 1 Tf
1.0555 0 TD
(O)Tj
/F9 1 Tf
0.7906 1.11 TD
( )Tj
/F2 1 Tf
2.765 -0.4335 TD
(1)Tj
ET
1 i
428.25 558.93 50.362 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 428.3069 548.8877 Tm
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 436.8847 547.2518 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 441.4496 548.8877 Tm
[(\(1\))-166.6(log)]TJ
/F8 1 Tf
2.9027 0 TD
(z)Tj
/F9 1 Tf
0.6186 1.796 TD
(\012)Tj
/F8 1 Tf
0.5972 -1.11 TD
(.)Tj
/F16 1 Tf
-31.6857 -4.0358 TD
[(Pr)51.2(o)51.1(o)0(f)]TJ
/F2 1 Tf
2.3267 0 TD
[(.)-819.3(Let)]TJ
/F8 1 Tf
2.7844 0 TD
(E)Tj
/F1 1 Tf
7.9701 0 0 7.9701 204.527 510.7079 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 209.2593 512.3437 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
0.2778 Tc
[(\):)277.8(=)]TJ
/F9 1 Tf
2 0.75 TD
0 Tc
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 253.08 509.1188 Tm
(d>z)Tj
/F10 1 Tf
4.9813 0 0 4.9813 268.2999 511.3883 Tm
(u)Tj
/F8 1 Tf
10.9091 0 0 10.9091 275.3285 512.3437 Tm
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 280.532 510.7079 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 285.0969 512.3437 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/d)Tj
/F2 1 Tf
1.0205 0 TD
[(;)-263.8(and)-228.9(note)-229(that)]TJ
/F7 1 Tf
6.5618 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F1 1 Tf
7.9701 0 0 7.9701 393.059 510.7079 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 397.7913 512.3437 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
0.2777 Tc
(|)Tj
/F9 1 Tf
1.6111 0.75 TD
0 Tc
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 441.6119 509.1188 Tm
(d)Tj
/F7 1 Tf
10.9091 0 0 10.9091 448.2859 512.3437 Tm
(|)Tj
/F8 1 Tf
0.2777 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 456.519 510.7079 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 461.084 512.3437 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(/d)Tj
/F7 1 Tf
-32.9521 -1.2694 TD
()Tj
/F2 1 Tf
1.2778 0 TD
(1)Tj
/F8 1 Tf
0.5 0 TD
(/G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 152.2251 496.8598 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 156.79 498.4956 Tm
[(\(1\).)-444.4(By)-333.4(a)-333.3(result)-333.3(of)-333.2(R.)-333.3(R.)-333.4(Hall)-333.2(\(see)-333.3([13],)-333.3(or)-333.3(\(4.1\))-333.3(of)-333.3([10]\))-333.3(w)27.8(e)-333.4(see)-333.3(that)]TJ
2.5646 -1.9729 TD
(1)Tj
ET
182.01 472.486 10.954 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 182.0174 462.1101 Tm
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 187.5706 465.2613 Tm
(u)Tj
/F9 1 Tf
10.9091 0 0 10.9091 198.0722 479.9572 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 195.9851 456.2401 Tm
(d)Tj
/F15 1 Tf
0.5467 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F10 1 Tf
4.9813 0 0 4.9813 211.205 458.5096 Tm
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 217.7355 469.593 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 225.9696 467.9572 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 230.5345 469.593 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
0.2777 Tc
(|)Tj
/F2 1 Tf
3.0461 0.6765 TD
0 Tc
(1)Tj
ET
265.89 472.486 29.525 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 265.8919 462.1101 Tm
(u)Tj
/F2 1 Tf
0.7392 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.7852 1.636 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 304.1315 456.2401 Tm
(d)Tj
/F7 1 Tf
10.9091 0 0 10.9091 317.2025 477.0341 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 325.4366 475.3973 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 330.0015 477.0341 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
(|)Tj
ET
317.25 472.486 29.992 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 329.3589 462.1101 Tm
(d)Tj
/F7 1 Tf
2.0223 0.6859 TD
()Tj
/F2 1 Tf
3.7318 0.6765 TD
(1)Tj
ET
366.57 472.486 56.607 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 366.5544 462.1101 Tm
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 375.1323 460.4732 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 379.6972 462.1101 Tm
(\(1\))Tj
/F8 1 Tf
1.2778 0 TD
(u)Tj
/F2 1 Tf
0.7391 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
0.6186 0.6859 TD
(.)Tj
/F2 1 Tf
-28.0095 -3.1226 TD
[(Therefore,)-333.2(from)-333.2(\(3.1\))-333.3(w)27.8(e)-333.4(deduce)-333.3(that)]TJ
/F8 1 Tf
6.1995 -2.4963 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
0.1667 Tc
(\)=)Tj
/F7 1 Tf
1.5001 0 TD
0 Tc
()Tj
/F2 1 Tf
0.7778 0 TD
[(\(1)-222.1(+)]TJ
/F8 1 Tf
2.111 0 TD
(O)Tj
/F2 1 Tf
0.7906 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(z)Tj
/F15 1 Tf
7.9701 0 0 7.9701 271.8984 412.801 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
(u)Tj
/F2 1 Tf
10.9091 0 0 10.9091 283.8864 408.2969 Tm
(\)\))Tj
/F8 1 Tf
0.7777 0 TD
(E)Tj
/F1 1 Tf
7.9701 0 0 7.9701 300.4244 406.66 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 305.1567 408.2969 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
0.2223 Tc
(\)+)Tj
/F8 1 Tf
1.6111 0 TD
0 Tc
(O)Tj
/F9 1 Tf
0.7906 1.11 TD
( )Tj
/F2 1 Tf
0.9588 -0.4334 TD
(1)Tj
ET
349.53 411.286 10.954 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 349.5542 400.8139 Tm
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 355.1074 403.9651 Tm
(u)Tj
/F9 1 Tf
10.9091 0 0 10.9091 365.609 418.66 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 363.5218 394.944 Tm
(d)Tj
/F15 1 Tf
0.5468 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F10 1 Tf
4.9813 0 0 4.9813 378.7417 397.2125 Tm
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 385.2722 408.2969 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 393.5064 406.66 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 398.0713 408.2969 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F9 1 Tf
0.2777 1.11 TD
(\012)Tj
/F2 1 Tf
-27.1759 -1.11 TD
(\(3.7\))Tj
8.5122 -3.1335 TD
(=)Tj
/F7 1 Tf
0.9445 0 TD
()Tj
/F8 1 Tf
0.7778 0 TD
(E)Tj
/F1 1 Tf
7.9701 0 0 7.9701 238.5016 372.4772 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 243.2339 374.1131 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
0.2222 Tc
(\)+)Tj
/F8 1 Tf
1.6111 0 TD
0 Tc
(O)Tj
/F9 1 Tf
0.7906 1.11 TD
( )Tj
/F2 1 Tf
3.0513 -0.4335 TD
(1)Tj
ET
287.61 377.086 56.607 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 287.6324 366.6301 Tm
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 296.2092 364.9943 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 300.7741 366.6301 Tm
(\(1\))Tj
/F8 1 Tf
1.2778 0 TD
(u)Tj
/F2 1 Tf
0.7391 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.6186 1.796 TD
(\012)Tj
/F8 1 Tf
0.5972 -1.11 TD
(.)Tj
/F2 1 Tf
-21.3721 -2.6533 TD
[(Manipulation)-244.8(of)]TJ
/F8 1 Tf
7.1565 0 TD
(E)Tj
/F1 1 Tf
7.9701 0 0 7.9701 204.9235 343.5317 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 209.6558 345.1676 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
[(\))-244.9(yields)-244.9(our)-244.9(iden)27.8(tit)27.8(y)83.3(.)-414.9(The)-245(starting)-244.8(p)-27.8(oin)27.8(t)-244.9(i)0(s)-244.9(the)-244.9(observ)55.6(ation)]TJ
-9.2899 -1.2694 TD
(that)Tj
/F8 1 Tf
0.5075 -3.7181 TD
(uE)Tj
/F1 1 Tf
7.9701 0 0 7.9701 138.6331 289.1218 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 143.3654 290.7586 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
0.2222 Tc
(\)+)Tj
/F9 1 Tf
1.6111 1.3611 TD
0 Tc
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 182.3382 302.4518 Tm
()Tj
/F14 1 Tf
-0.6084 -2.7082 TD
(u)Tj
/F8 1 Tf
10.9091 0 0 10.9091 193.1228 290.7586 Tm
(E)Tj
/F1 1 Tf
7.9701 0 0 7.9701 201.1756 289.1218 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 205.9079 290.7586 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(t)Tj
/F2 1 Tf
0.3612 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(dt)Tj
/F2 1 Tf
1.1594 0 TD
(=)Tj
/F8 1 Tf
1.0555 0 TD
(uE)Tj
/F1 1 Tf
7.9701 0 0 7.9701 256.793 289.1218 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 261.5253 290.7586 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
0.2222 Tc
(\)+)Tj
/F9 1 Tf
1.8025 0.95 TD
0 Tc
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 289.588 277.4058 Tm
(d>z)Tj
/F10 1 Tf
4.9813 0 0 4.9813 304.808 279.6752 Tm
(u)Tj
/F8 1 Tf
10.9091 0 0 10.9091 312.534 298.1987 Tm
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 317.7375 296.5629 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 322.3024 298.1987 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
ET
312.57 293.686 23.931 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 321.6608 283.2757 Tm
(d)Tj
/F9 1 Tf
1.4667 1.7959 TD
( )Tj
/F2 1 Tf
0.7069 -0.4334 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(d)Tj
ET
345.33 293.686 21.587 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 345.4346 283.2757 Tm
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F7 1 Tf
0.8466 0.6859 TD
()Tj
/F8 1 Tf
1 0 TD
(u)Tj
/F9 1 Tf
0.5725 1.11 TD
(\012)Tj
/F2 1 Tf
0.875 -1.11 TD
(=)Tj
/F9 1 Tf
1.2469 0.95 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 408.793 277.4058 Tm
(d>z)Tj
/F10 1 Tf
4.9813 0 0 4.9813 424.013 279.6752 Tm
(u)Tj
/F8 1 Tf
10.9091 0 0 10.9091 431.739 298.1987 Tm
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 436.9425 296.5629 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 441.5074 298.1987 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
ET
431.73 293.686 23.931 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 440.8658 283.2757 Tm
(d)Tj
/F2 1 Tf
1.5763 1.3625 TD
(log)Tj
/F8 1 Tf
1.4583 0 TD
(d)Tj
ET
458.01 293.686 21.587 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 458.123 283.2757 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
0.6244 0.6859 TD
(.)Tj
/F2 1 Tf
-33.1874 1.9042 TD
(\(3.8\))Tj
0 -4.9497 TD
[(W)83.3(e)-333.3(appro)27.8(ximate)-333.3(the)-333.3(left)-333.3(side)-333.3(of)-333.3(\(3.8\))-333.3(as)-333.4(follo)27.8(ws:)]TJ
/F9 1 Tf
2.8953 -1.4119 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F2 1 Tf
0.3333 0.0501 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(uE)Tj
/F1 1 Tf
7.9701 0 0 7.9701 172.5619 227.9511 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 177.2941 229.5881 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
0.2223 Tc
(\)+)Tj
/F9 1 Tf
1.6111 1.3611 TD
0 Tc
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 216.266 241.2812 Tm
()Tj
/F14 1 Tf
-0.6084 -2.7083 TD
(u)Tj
/F8 1 Tf
10.9091 0 0 10.9091 227.0506 229.5881 Tm
(E)Tj
/F1 1 Tf
7.9701 0 0 7.9701 235.1034 227.9511 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 239.8356 229.5881 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(t)Tj
/F2 1 Tf
0.3611 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(dt)Tj
/F2 1 Tf
0.8816 0 TD
0.2222 Tc
(\)+\()Tj
/F8 1 Tf
2 0 TD
0 Tc
(uE)Tj
/F2 1 Tf
1.3682 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
0.2223 Tc
(\)+)Tj
/F9 1 Tf
1.6111 1.3611 TD
0 Tc
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 337.595 241.2812 Tm
()Tj
/F14 1 Tf
-0.6084 -2.7083 TD
(u)Tj
/F8 1 Tf
10.9091 0 0 10.9091 348.3796 229.5881 Tm
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(t)Tj
/F2 1 Tf
0.3611 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(dt)Tj
/F2 1 Tf
0.8816 0 TD
(\))Tj
/F9 1 Tf
0.3889 1.1499 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F7 1 Tf
-17.1629 -2.6085 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(u)Tj
/F7 1 Tf
0.5724 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F1 1 Tf
7.9701 0 0 7.9701 224.9564 198.9489 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 229.6887 200.5858 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
0.2222 Tc
(\)+)Tj
/F8 1 Tf
1.6111 0 TD
0 Tc
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F2 1 Tf
0.5 0 TD
(+)Tj
/F9 1 Tf
1 1.3611 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 308.4364 212.2789 Tm
()Tj
/F14 1 Tf
-0.6084 -2.7084 TD
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 319.2209 200.5858 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F1 1 Tf
7.9701 0 0 7.9701 330.3044 198.9489 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 335.0366 200.5858 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(t)Tj
/F2 1 Tf
0.3612 0 TD
0.2222 Tc
(\)+)Tj
/F8 1 Tf
1.6111 0 TD
0 Tc
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(t)Tj
/F2 1 Tf
0.3611 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
(|)Tj
/F8 1 Tf
0.2777 0 TD
(dt)Tj
/F7 1 Tf
-17.3082 -2.6586 TD
()Tj
/F2 1 Tf
3.4456 0.6765 TD
(1)Tj
ET
211.29 174.526 50.362 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 211.2478 164.0996 Tm
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 219.8257 162.4637 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 224.3905 164.0996 Tm
[(\(1\))-166.6(log)]TJ
/F8 1 Tf
2.9027 0 TD
(z)Tj
/F2 1 Tf
0.8408 0.6859 TD
(+)Tj
/F9 1 Tf
1 1.3611 TD
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 287.0476 183.2767 Tm
()Tj
/F14 1 Tf
-0.6082 -2.7084 TD
(u)Tj
/F8 1 Tf
10.9091 0 0 10.9091 297.8321 171.5825 Tm
(z)Tj
/F15 1 Tf
7.9701 0 0 7.9701 303.3853 176.0866 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
(t)Tj
/F9 1 Tf
10.9091 0 0 10.9091 313.5283 183.6922 Tm
( )Tj
/F2 1 Tf
1.6981 -0.4335 TD
(1)Tj
ET
321.21 174.526 27.081 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 321.2384 164.0996 Tm
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 329.8162 162.4637 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 334.381 164.0996 Tm
(\(1\))Tj
1.6096 0.6859 TD
(+)Tj
/F9 1 Tf
1.1913 0.9501 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 362.8493 158.2296 Tm
(d)Tj
/F15 1 Tf
0.5466 0 TD
()Tj
/F14 1 Tf
0.8265 0 TD
(z)Tj
/F10 1 Tf
4.9813 0 0 4.9813 378.0683 160.4991 Tm
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 384.5998 171.5825 Tm
(|)Tj
/F8 1 Tf
0.2777 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 392.8329 169.9467 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 397.3978 171.5825 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F9 1 Tf
0.2778 1.11 TD
(\012)Tj
/F8 1 Tf
0.5973 -1.11 TD
(dt)Tj
/F7 1 Tf
-20.6244 -3.2446 TD
()Tj
/F2 1 Tf
3.4456 0.6765 TD
(1)Tj
ET
211.29 139.126 50.362 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 211.2478 128.7043 Tm
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 219.8257 127.0674 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 224.3905 128.7043 Tm
[(\(1\))-166.6(log)]TJ
/F8 1 Tf
2.9027 0 TD
(z)Tj
/F2 1 Tf
0.8408 0.6859 TD
(+)Tj
1.8433 0.6765 TD
(1)Tj
ET
277.29 139.126 21.463 -0.436 re
f
BT
10.9091 0 0 10.9091 277.334 128.7043 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.6186 1.796 TD
( )Tj
0.7638 -0.16 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 314.0254 122.8343 Tm
(d)Tj
/F7 1 Tf
10.9091 0 0 10.9091 327.0964 143.6283 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 335.3295 141.9915 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 339.8944 143.6283 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
ET
327.09 139.126 29.992 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 339.2528 128.7043 Tm
(d)Tj
/F9 1 Tf
1.7445 1.796 TD
(\012)Tj
/F7 1 Tf
0.875 -1.11 TD
()Tj
/F2 1 Tf
3.4457 0.6765 TD
(1)Tj
ET
383.01 138.69 50.362 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 382.9639 128.7043 Tm
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 391.5408 127.0674 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 396.1056 128.7043 Tm
[(\(1\))-166.7(log)]TJ
/F8 1 Tf
2.9028 0 TD
(z)Tj
0.6186 0.6859 TD
(.)Tj
ET
endstream
endobj
95 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F10 13 0 R
/F14 22 0 R
/F15 23 0 R
/F16 24 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
97 0 obj
<<
/Length 18952
>>
stream
BT
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
(24)Tj
/F5 1 Tf
7.9701 0 0 7.9701 202.119 726.9739 Tm
[(ANDREW)-377.5(GRANVILLE)-377.4(AND)-377.7(K)0(.)-377.7(SOUND)29.4(ARARAJAN)]TJ
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 701.071 Tm
[(No)27.8(w)-333.4(log)]TJ
/F8 1 Tf
3.7361 0 TD
(d)Tj
/F2 1 Tf
0.7983 0 TD
(=)Tj
/F9 1 Tf
1.0556 0.75 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 191.2956 697.8451 Tm
(m)Tj
/F15 1 Tf
0.9399 0 TD
(|)Tj
/F14 1 Tf
0.2951 0 TD
(d)Tj
/F2 1 Tf
10.9091 0 0 10.9091 207.8117 701.071 Tm
(\()Tj
/F8 1 Tf
1.0834 0 TD
(m)Tj
/F2 1 Tf
0.878 0 TD
[(\))-333.4(s)0(o)-333.3(that)-333.3(the)-333.3(righ)27.8(t)-333.3(side)-333.3(of)-333.3(\(3.8\))-333.3(equals)]TJ
2.132 -1.7408 TD
(1)Tj
ET
244.41 677.686 21.463 -0.436 re
f
BT
10.9091 0 0 10.9091 244.4633 667.217 Tm
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F9 1 Tf
0.7853 1.636 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 273.072 662.1549 Tm
(m)Tj
/F2 1 Tf
10.9091 0 0 10.9091 286.5146 674.7 Tm
(\()Tj
/F8 1 Tf
1.0833 0 TD
(m)Tj
/F2 1 Tf
0.8781 0 TD
(\))Tj
/F9 1 Tf
0.7408 0.95 TD
()Tj
/F10 1 Tf
4.9813 0 0 4.9813 315.1671 663.2708 Tm
-0.0002 Tc
(d>z)Tj
2.5488 0.2944 TD
0 Tc
(u)Tj
-2.0144 -1.6538 TD
(m)Tj
/F11 1 Tf
1.228 0 TD
(|)Tj
/F10 1 Tf
0.4584 0 TD
(d)Tj
/F8 1 Tf
10.9091 0 0 10.9091 336.7851 682.141 Tm
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 341.9886 680.5041 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 346.5534 682.141 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
ET
336.81 677.686 23.931 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 345.9118 667.217 Tm
(d)Tj
1.4667 0.6859 TD
(.)Tj
/F2 1 Tf
-22.2853 -3.2597 TD
[(The)-359.5(sum)-359.6(o)27.8(v)27.8(e)0(r)]TJ
/F8 1 Tf
6.3925 0 TD
(m)Tj
/F2 1 Tf
0.8781 0 TD
[(s)-359.5(ab)-27.7(o)27.8(v)27.8(e)-359.6(can)-359.5(b)-27.7(e)-359.6(restricted)-359.4(to)-359.6(prime)-359.4(p)-27.8(o)27.8(w)27.8(ers)]TJ
/F8 1 Tf
19.0709 0 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 411.6494 643.0984 Tm
(k)Tj
/F2 1 Tf
10.9091 0 0 10.9091 420.6914 639.1393 Tm
(for)Tj
/F8 1 Tf
1.5567 0 TD
(p)Tj
/F7 1 Tf
0.8247 0 TD
()Tj
/F8 1 Tf
1.0993 0 TD
(z)Tj
/F2 1 Tf
0.8685 0 TD
(\(else)Tj
/F8 1 Tf
-32.0226 -1.2694 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 124.003 623.6543 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 128.5679 625.2911 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
[(\))-277.7(=)-277.8(0\).)-426.3(F)83.4(urther)-278.9(the)-278.8(con)27.8(tribution)-278.8(of)-278.9(prime)-278.9(p)-27.7(o)27.8(w)27.8(ers)]TJ
/F8 1 Tf
21.3276 0 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 376.6416 629.2504 Tm
(k)Tj
/F2 1 Tf
10.9091 0 0 10.9091 384.804 625.2911 Tm
(with)Tj
/F8 1 Tf
2.2234 0 TD
(k)Tj
/F7 1 Tf
0.8299 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
[(2)-279(i)0(s)-278.9(b)-27.7(ounded)]TJ
-28.4925 -1.2694 TD
-0.0278 Tc
(by)Tj
/F7 1 Tf
1.3889 0 TD
0 Tc
()Tj
/F2 1 Tf
1.2778 0 TD
(1)Tj
/F8 1 Tf
0.5 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 171.6194 609.8062 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 176.1843 611.4431 Tm
[(\(1\))-166.6(log)]TJ
/F8 1 Tf
2.9027 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
[(\).)-444.5(Finally)-333.2(for)]TJ
/F8 1 Tf
6.0445 0 TD
(m)Tj
/F2 1 Tf
1.1558 0 TD
(=)Tj
/F8 1 Tf
1.0555 0 TD
(p)Tj
/F7 1 Tf
0.7809 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(z)Tj
/F2 1 Tf
0.8424 0 TD
[(w)27.8(e)-333.4(see)-333.3(that)]TJ
/F9 1 Tf
-14.3776 -1.2811 TD
()Tj
/F10 1 Tf
4.9813 0 0 4.9813 175.0187 575.6743 Tm
-0.0002 Tc
(d>z)Tj
2.5486 0.2946 TD
0 Tc
(u)Tj
-2.0144 -1.654 TD
(m)Tj
/F11 1 Tf
1.2282 0 TD
(|)Tj
/F10 1 Tf
0.4582 0 TD
(d)Tj
/F8 1 Tf
10.9091 0 0 10.9091 196.6366 594.5444 Tm
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 201.8391 592.9086 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 206.404 594.5444 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
ET
196.65 590.086 23.931 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 205.7624 579.6215 Tm
(d)Tj
/F2 1 Tf
1.6334 0.6859 TD
(=)Tj
/F8 1 Tf
1.0541 0.682 TD
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 240.2839 592.9086 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 244.8488 594.5444 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
235.05 590.086 23.742 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 244.2072 579.6215 Tm
(p)Tj
/F9 1 Tf
2.1967 1.6359 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 261.8361 573.3081 Tm
(d>z)Tj
/F10 1 Tf
4.9813 0 0 4.9813 277.0551 575.5776 Tm
(u)Tj
/F14 1 Tf
7.9701 0 0 7.9701 281.7684 573.3081 Tm
(/p)Tj
/F8 1 Tf
10.9091 0 0 10.9091 293.2782 594.5444 Tm
(g)Tj
/F14 1 Tf
7.9701 0 0 7.9701 298.4817 592.9086 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 303.0466 594.5444 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\))Tj
ET
293.25 590.086 23.931 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 302.405 579.6215 Tm
(d)Tj
/F2 1 Tf
1.6889 0.6859 TD
(+)Tj
/F8 1 Tf
1 0 TD
(O)Tj
/F9 1 Tf
0.7906 1.11 TD
( )Tj
/F2 1 Tf
2.1665 -0.4334 TD
(1)Tj
ET
1 i
348.09 589.65 37.302 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 348.0737 579.6215 Tm
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 353.5622 582.7726 Tm
(2)Tj
/F8 1 Tf
10.9091 0 0 10.9091 358.2944 579.6215 Tm
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 366.8712 577.9846 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 371.4361 579.6215 Tm
(\(1\))Tj
/F9 1 Tf
1.3874 1.7959 TD
(\012)Tj
/F2 1 Tf
-14.9407 -4.79 TD
(=)Tj
/F7 1 Tf
0.9445 0 TD
()Tj
/F2 1 Tf
0.8874 0.6821 TD
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F8 1 Tf
1 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 267.6942 552.7632 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 272.259 554.4 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
243.57 549.886 42.667 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 262.1549 539.476 Tm
(p)Tj
2.3167 0.6859 TD
(E)Tj
/F1 1 Tf
7.9701 0 0 7.9701 295.481 545.3231 Tm
(1)Tj
/F9 1 Tf
10.9091 0 0 10.9091 300.2132 559.0685 Tm
( )Tj
/F8 1 Tf
0.5973 -1.11 TD
(u)Tj
/F7 1 Tf
0.7946 0 TD
()Tj
/F2 1 Tf
1.1126 0.6765 TD
(log)Tj
/F8 1 Tf
1.4583 0 TD
(p)Tj
ET
327.45 549.45 21.462 0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 327.5019 539.476 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.6186 1.796 TD
(\012)Tj
/F2 1 Tf
0.8195 -1.11 TD
(+)Tj
/F8 1 Tf
1 0 TD
(O)Tj
/F9 1 Tf
0.7905 1.11 TD
( )Tj
/F2 1 Tf
2.1666 -0.4335 TD
(1)Tj
ET
386.37 549.886 37.302 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 386.3432 539.476 Tm
(p)Tj
/F1 1 Tf
7.9701 0 0 7.9701 391.8316 542.6282 Tm
(2)Tj
/F8 1 Tf
10.9091 0 0 10.9091 396.5639 539.476 Tm
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 405.1417 537.8401 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 409.7066 539.476 Tm
(\(1\))Tj
/F9 1 Tf
1.3873 1.796 TD
(\012)Tj
/F8 1 Tf
0.5973 -1.11 TD
(.)Tj
/F2 1 Tf
-28.6511 -2.2349 TD
[(Therefore,)-485(b)27.8(y)-454.8(\(3.7\),)-485(this,)-485.1(tak)27.8(en)-454.8(with)-454.7(the)-454.7(estimate)-454.7(for)-454.7(the)-454.8(left)-454.7(side)-454.7(of)-454.8(\(3.8\),)]TJ
0 -1.2694 TD
[(yields)-333.3(the)-333.3(result.)]TJ
2.0091 -1.726 TD
[(W)83.3(e)-358.4(call)-358.3(a)-358.4(p)-27.8(oin)27.8(t)]TJ
/F8 1 Tf
7.1002 0 TD
(w)Tj
/F16 1 Tf
1.1012 0 TD
[(sp)51.1(e)51.1(cial)]TJ
/F2 1 Tf
3.1695 0 TD
(if)Tj
/F7 1 Tf
0.9417 0 TD
(|)Tj
/F8 1 Tf
0.2777 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F2 1 Tf
0.5974 0 TD
[(=)-319.5(max)]TJ
/F14 1 Tf
7.9701 0 0 7.9701 342.126 488.264 Tm
(x)Tj
/F15 1 Tf
0.5981 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(w)Tj
/F7 1 Tf
10.9091 0 0 10.9091 362.0872 489.9009 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F2 1 Tf
0.2778 0 TD
[(.)-519.6(Since)]TJ
/F8 1 Tf
3.4335 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F7 1 Tf
0.7084 0 TD
( )Tj
/F2 1 Tf
1.3195 0 TD
0.3584 Tc
[(0a)358.4(s)]TJ
/F8 1 Tf
-32.2198 -1.2694 TD
0 Tc
(x)Tj
/F7 1 Tf
0.8493 0 TD
0.2778 Tc
( )Tj
/F2 1 Tf
2.6111 0 TD
0 Tc
[(w)27.8(e)-333.3(see)-333.4(that)-333.3(there)-333.3(are)-333.3(arbitrarily)-333.2(large)-333.3(sp)-27.8(ecial)-333.3(p)-27.7(oin)27.8(ts.)]TJ
/F5 1 Tf
-1.4513 -1.8174 TD
[(Lemma)-325.8(3.10.)]TJ
/F16 1 Tf
6.3987 0 TD
(Given)Tj
/F8 1 Tf
2.8748 0 TD
(u)Tj
/F7 1 Tf
0.8502 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(2)Tj
/F16 1 Tf
0.8123 0 TD
(either)Tj
/F8 1 Tf
2.8039 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5716 0 TD
0.2778 Tc
(\)=)Tj
/F8 1 Tf
1.7222 0 TD
0 Tc
(O)Tj
/F2 1 Tf
0.7905 0 TD
(\(1)Tj
/F8 1 Tf
0.889 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 376.6655 454.5903 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 381.2304 456.2271 Tm
[(\(1\))-166.6(log)]TJ
/F8 1 Tf
2.9027 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\)\))Tj
/F16 1 Tf
1.09 0 TD
[(for)-312.2(al)-51.1(l)]TJ
/F8 1 Tf
2.9374 0 TD
(x)Tj
/F7 1 Tf
0.8492 0 TD
()Tj
/F8 1 Tf
1.0556 0 TD
(u)Tj
/F16 1 Tf
-33.4002 -1.2694 TD
[(or)-357.8(ther)51.2(e)-357.8(i)0(s)-357.8(a)-357.7(sp)51.1(e)51.1(cial)-357.7(p)51.1(oint)-357.8(in)]TJ
/F2 1 Tf
12.6499 0 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
[(u,)-166.6(u)]TJ
/F2 1 Tf
1.8116 0 TD
0.2222 Tc
[(+1)222.2(])]TJ
/F16 1 Tf
1.7778 0 TD
0 Tc
(.)Tj
-14.5079 -1.8174 TD
[(Pr)51.2(o)51.1(of.)]TJ
/F2 1 Tf
3.4991 0 TD
(Let)Tj
/F8 1 Tf
1.9321 0 TD
(w)Tj
/F2 1 Tf
1.2166 0 TD
[(denote)-473.7(the)-473.7(inm)27.7(um)-473.7(of)-473.8(the)-473.7(set)-473.8(of)-473.8(sp)-27.8(ecial)-473.7(p)-27.8(oin)27.8(ts)-473.7(at)-473.7(least)]TJ
-8.657 -1.2694 TD
[(as)-452.8(large)-452.7(as)]TJ
/F8 1 Tf
5.2613 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
[(,)-482.8(and)-452.8(assume)]TJ
/F8 1 Tf
6.3995 0 TD
0.5038 Tc
[(w>)26.9(u)]TJ
/F2 1 Tf
3.3489 0 TD
0 Tc
[(+)-301.9(1)-452.8(\(that)-452.8(is,)-482.7(there)-452.8(is)-452.8(no)-452.8(sp)-27.8(ecial)-452.8(p)-27.7(oin)27.8(t)-452.8(i)0(n)]TJ
-15.5821 -1.2694 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
[(u,)-166.6(u)]TJ
/F2 1 Tf
1.9056 0 TD
[(+)-316.3(1)0(]\).)-867.7(Note)-474.3(that)]TJ
/F7 1 Tf
8.2715 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
0.5129 Tc
(||)Tj
/F8 1 Tf
2.3591 0 TD
0 Tc
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5716 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F2 1 Tf
0.5941 0 TD
(+)Tj
/F8 1 Tf
1.094 0 TD
(O)Tj
/F2 1 Tf
0.7906 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(z)Tj
/F15 1 Tf
7.9701 0 0 7.9701 347.1253 398.8164 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
(u)Tj
/F2 1 Tf
10.9091 0 0 10.9091 359.1123 394.8573 Tm
[(\))-474.4(for)-474.4(an)27.7(y)]TJ
/F8 1 Tf
4.5648 0 TD
(x)Tj
/F7 1 Tf
1.0844 0 TD
()Tj
/F8 1 Tf
1.2907 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
0.8676 Tc
[(.I)867.6(f)]TJ
/F8 1 Tf
2.2865 0 TD
0 Tc
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5716 0 TD
(\))Tj
-33.5838 -1.2694 TD
[(main)27.8(tains)-373.2(the)-373.3(same)-373.2(sign)-373.3(for)-373.3(all)]TJ
/F8 1 Tf
14.037 0 TD
(x)Tj
/F7 1 Tf
0.916 0 TD
()Tj
/F8 1 Tf
1.1221 0 TD
(w)Tj
/F2 1 Tf
1.1162 0 TD
(set)Tj
/F8 1 Tf
1.6011 0 TD
(v)Tj
/F2 1 Tf
0.8649 0 TD
(=)Tj
/F7 1 Tf
1.1222 0 TD
()Tj
/F2 1 Tf
1 0 TD
[(;)-393.3(otherwise)-373.1(let)]TJ
/F8 1 Tf
6.6482 0 TD
(v)Tj
/F2 1 Tf
0.8939 0 TD
[(denote)-373.2(the)]TJ
-29.3216 -1.2694 TD
[(inm)27.8(um)-338.4(of)-338.4(those)-338.3(p)-27.8(oin)27.8(ts)]TJ
/F8 1 Tf
10.6981 0 TD
(x)Tj
/F7 1 Tf
0.8577 0 TD
()Tj
/F8 1 Tf
1.064 0 TD
(w)Tj
/F2 1 Tf
1.0812 0 TD
[(for)-338.4(whic)27.8(h)]TJ
/F8 1 Tf
4.4018 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
[(\))-338.4(has)-338.4(the)-338.4(opp)-27.8(osite)-338.3(sign)-338.4(to)]TJ
/F8 1 Tf
11.5193 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\).)Tj
-33.3059 -1.2694 TD
[(Note)-407.1(that)]TJ
/F8 1 Tf
4.731 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(v)Tj
/F2 1 Tf
0.5206 0 TD
0.4008 Tc
(\)=)Tj
/F8 1 Tf
1.9682 0 TD
0 Tc
(O)Tj
/F2 1 Tf
0.7905 0 TD
(\(1)Tj
/F8 1 Tf
0.8889 0 TD
(/z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 239.8127 357.2722 Tm
(v)Tj
/F2 1 Tf
10.9091 0 0 10.9091 244.7203 353.313 Tm
[(\).)-665.8(T)83.3(aking)-407.1(Lemma)-407(3.9)-407.1(with)]TJ
/F8 1 Tf
12.4194 0 TD
(u)Tj
/F2 1 Tf
0.9732 0 TD
(=)Tj
/F8 1 Tf
1.1785 0 TD
(w)Tj
/F2 1 Tf
1.1499 0 TD
(and)Tj
/F8 1 Tf
2.0183 0 TD
(u)Tj
/F2 1 Tf
0.9732 0 TD
(=)Tj
/F8 1 Tf
1.1785 0 TD
(v)Tj
/F2 1 Tf
0.9277 0 TD
(and)Tj
-32.3615 -1.2694 TD
[(subtracting)-333.2(w)27.8(e)-333.4(nd)-333.3(that)]TJ
0 -1.8639 TD
(\(3.9\))Tj
/F8 1 Tf
0.9132 -1.9042 TD
0.0269 Tc
(wE)Tj
/F2 1 Tf
1.5386 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
0.0402 Tc
(\)+)Tj
/F9 1 Tf
1.247 1.3611 TD
0 Tc
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 182.407 310.0513 Tm
(v)Tj
-0.6084 -2.7083 TD
(w)Tj
/F8 1 Tf
10.9091 0 0 10.9091 189.1328 298.3582 Tm
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(t)Tj
/F2 1 Tf
0.3611 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(dt)Tj
/F2 1 Tf
0.9217 0 TD
(+)Tj
1.6613 0.6765 TD
(1)Tj
ET
230.37 301.246 21.463 -0.436 re
f
BT
10.9091 0 0 10.9091 230.413 290.8752 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.7852 1.6359 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 255.205 285.2104 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 273.6598 305.7982 Tm
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F8 1 Tf
1 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 297.7883 304.1624 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 302.3532 305.7982 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
273.69 301.246 42.667 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 292.2491 290.8752 Tm
(p)Tj
/F2 1 Tf
2.4834 0.6859 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(p)Tj
/F9 1 Tf
0.7454 1.11 TD
( )Tj
/F8 1 Tf
0.5973 -1.11 TD
(E)Tj
/F9 1 Tf
0.7958 1.11 TD
( )Tj
/F8 1 Tf
0.5973 -1.11 TD
(w)Tj
/F7 1 Tf
0.783 0 TD
()Tj
/F2 1 Tf
0.9305 0.6765 TD
(log)Tj
/F8 1 Tf
1.4583 0 TD
(p)Tj
ET
383.73 301.246 21.462 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 383.7539 290.8752 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.6186 1.7959 TD
(\012)Tj
/F7 1 Tf
0.6374 -1.11 TD
()Tj
/F8 1 Tf
0.8179 0 TD
(E)Tj
/F9 1 Tf
0.7959 1.11 TD
( )Tj
/F8 1 Tf
0.5972 -1.11 TD
(v)Tj
/F7 1 Tf
0.5608 0 TD
()Tj
/F2 1 Tf
0.9305 0.6765 TD
(log)Tj
/F8 1 Tf
1.4583 0 TD
(p)Tj
ET
453.69 301.246 21.463 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 453.7216 290.8752 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.6186 1.7959 TD
(\012\012)Tj
/F2 1 Tf
-7.8728 -4.2435 TD
(=)Tj
/F8 1 Tf
1.0555 0 TD
(O)Tj
/F9 1 Tf
0.7906 1.11 TD
( )Tj
/F2 1 Tf
2.765 -0.4335 TD
(1)Tj
ET
418.29 266.73 50.362 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 418.3442 256.6914 Tm
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 426.922 255.0555 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 431.4869 256.6914 Tm
[(\(1\))-166.6(log)]TJ
/F8 1 Tf
2.9027 0 TD
(z)Tj
/F9 1 Tf
0.6186 1.796 TD
(\012)Tj
/F8 1 Tf
0.5972 -1.11 TD
(.)Tj
/F2 1 Tf
-32.7816 -2.2349 TD
(Since)Tj
/F8 1 Tf
2.6111 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(t)Tj
/F2 1 Tf
0.3612 0 TD
[(\))-333.3(main)27.8(tains)-333.2(the)-333.4(same)-333.3(sign)-333.3(throughout)-333.2([)]TJ
/F8 1 Tf
17.1027 0 TD
0.0269 Tc
[(w,)-139.8(v)]TJ
/F2 1 Tf
1.7079 0 TD
0 Tc
[(])-333.4(w)27.8(e)-333.4(h)0(a)27.8(v)27.8(e)-333.4(that)]TJ
/F9 1 Tf
-13.2847 -0.9934 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F8 1 Tf
0.3333 0.05 TD
0.0269 Tc
(wE)Tj
/F2 1 Tf
1.5386 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.389 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
0.2222 Tc
(\)+)Tj
/F9 1 Tf
1.611 1.3612 TD
0 Tc
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 285.6827 228.1046 Tm
(v)Tj
-0.6082 -2.7084 TD
(w)Tj
/F8 1 Tf
10.9091 0 0 10.9091 292.4085 216.4105 Tm
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(t)Tj
/F2 1 Tf
0.3611 0 TD
(\))Tj
/F8 1 Tf
0.389 0 TD
(dt)Tj
/F9 1 Tf
0.8816 1.15 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F7 1 Tf
0.6111 0.05 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(w)Tj
/F7 1 Tf
0.7428 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F8 1 Tf
0.2777 0 TD
(,)Tj
/F2 1 Tf
-24.012 -2.1081 TD
[(while)-333.3(on)-333.3(the)-333.3(other)-333.4(hand)]TJ
/F9 1 Tf
0.3334 -0.789 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F2 1 Tf
1.1766 0.7266 TD
(1)Tj
ET
127.29 175.246 21.463 -0.436 re
f
BT
10.9091 0 0 10.9091 127.2687 164.777 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.7852 1.6359 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 152.0608 159.1123 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 170.5156 179.7001 Tm
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F8 1 Tf
1 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 194.6441 178.0642 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 199.209 179.7001 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
170.49 175.246 42.667 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 189.1049 164.777 Tm
(p)Tj
/F2 1 Tf
2.4834 0.6859 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(p)Tj
/F9 1 Tf
0.8364 1.11 TD
( )Tj
/F8 1 Tf
0.5973 -1.11 TD
(E)Tj
/F9 1 Tf
0.7958 1.11 TD
( )Tj
/F8 1 Tf
0.5973 -1.11 TD
(w)Tj
/F7 1 Tf
0.965 0 TD
()Tj
/F2 1 Tf
1.1125 0.6765 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(p)Tj
ET
285.57 175.246 21.463 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 285.5751 164.777 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.6185 1.7959 TD
(\012)Tj
/F7 1 Tf
0.8195 -1.11 TD
()Tj
/F8 1 Tf
1 0 TD
(E)Tj
/F9 1 Tf
0.7959 1.11 TD
( )Tj
/F8 1 Tf
0.5972 -1.11 TD
(v)Tj
/F7 1 Tf
0.7428 0 TD
()Tj
/F2 1 Tf
1.1125 0.6765 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(p)Tj
ET
363.45 175.246 21.463 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 363.486 164.777 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.6186 1.7959 TD
-0.0001 Tc
(\012\012)Tj
1.1944 0.04 TD
0 Tc
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F7 1 Tf
-23.3676 -3.1334 TD
()Tj
/F2 1 Tf
1.8988 0.6765 TD
(2)Tj
ET
156.93 140.446 21.463 -0.436 re
f
BT
10.9091 0 0 10.9091 156.9654 130.0483 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.7853 1.636 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 181.7574 124.3836 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 200.2122 144.9115 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(p)Tj
ET
200.25 140.446 21.398 -0.436 re
f
BT
10.9091 0 0 10.9091 208.1674 130.0483 Tm
(p)Tj
/F2 1 Tf
1.849 0.6859 TD
(max)Tj
/F14 1 Tf
7.9701 0 0 7.9701 224.6237 130.1777 Tm
()Tj
/F15 1 Tf
0.5061 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(w)Tj
/F15 1 Tf
0.7894 0 TD
()Tj
/F1 1 Tf
0.8264 0 TD
(1)Tj
/F7 1 Tf
10.9091 0 0 10.9091 254.1738 137.5313 Tm
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
0.2778 Tc
(|)Tj
/F2 1 Tf
1.6111 0 TD
0 Tc
[(\(2)-222.2(+)]TJ
/F8 1 Tf
2.1111 0 TD
(o)Tj
/F2 1 Tf
0.4847 0 TD
[(\(1\)\))-166.6(max)]TJ
/F14 1 Tf
7.9701 0 0 7.9701 347.9293 130.1777 Tm
()Tj
/F15 1 Tf
0.5061 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(u)Tj
/F7 1 Tf
10.9091 0 0 10.9091 367.6613 137.5313 Tm
(|)Tj
/F8 1 Tf
0.2777 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
0.2777 Tc
(|)Tj
/F2 1 Tf
1.611 0 TD
0 Tc
[(\(2)-222.2(+)]TJ
/F8 1 Tf
2.1111 0 TD
(o)Tj
/F2 1 Tf
0.4847 0 TD
(\(1\)\))Tj
/F7 1 Tf
1.6667 0 TD
(|)Tj
/F8 1 Tf
0.2777 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(,)Tj
ET
endstream
endobj
98 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F10 13 0 R
/F11 14 0 R
/F14 22 0 R
/F15 23 0 R
/F16 24 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
100 0 obj
<<
/Length 18436
>>
stream
BT
/F5 1 Tf
7.9701 0 0 7.9701 162.3741 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
[(AN)-377.7(UNCER)87.7(T)87.5(AINTY)-377.5(PRINCIPLE)-377.6(F)29.2(O)0(R)-377.8(ARITHMETIC)-377.4(SEQUENCES)]TJ
/F2 1 Tf
10.9091 0 0 10.9091 478.5006 726.9741 Tm
(25)Tj
-32.9726 -2.3745 TD
(since)Tj
/F8 1 Tf
2.45 0 TD
(w)Tj
/F2 1 Tf
1.0762 0 TD
[(w)27.8(a)0(s)-333.3(assumed)-333.3(to)-333.3(b)-27.7(e)-333.4(larger)-333.3(than)]TJ
/F8 1 Tf
13.6889 0 TD
(u)Tj
/F2 1 Tf
0.7947 0 TD
[(+)-222.2(1)0(.)-444.4(W)83.3(e)-333.3(conclude)-333.3(that)]TJ
-10.136 -1.664 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(u)Tj
/F7 1 Tf
0.7947 0 TD
()Tj
/F2 1 Tf
1 0 TD
0.2222 Tc
(1+)Tj
/F8 1 Tf
1.7222 0 TD
0 Tc
(o)Tj
/F2 1 Tf
0.4847 0 TD
(\(1\)\))Tj
/F7 1 Tf
1.6667 0 TD
(|)Tj
/F8 1 Tf
0.2777 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F2 1 Tf
0.5556 0 TD
(=)Tj
/F8 1 Tf
1.0555 0 TD
(O)Tj
/F2 1 Tf
0.7905 0 TD
(\(1)Tj
/F8 1 Tf
0.8889 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 353.2453 681.2822 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 357.8102 682.9191 Tm
[(\(1\))-166.6(log)]TJ
/F8 1 Tf
2.9027 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\)\))Tj
-25.3211 -1.664 TD
[(whic)27.8(h)-333.3(establishes)-333.2(the)-333.4(lemma.)]TJ
ET
482.61 671.529 6.775 -0.399 re
f
482.61 671.108 0.399 -5.978 re
f
488.97 671.108 0.399 -5.978 re
f
482.61 665.169 6.775 -0.399 re
f
BT
/F5 1 Tf
10.9091 0 0 10.9091 140.7173 640.9554 Tm
[(Lemma)-450.7(3.11)]TJ
/F2 1 Tf
5.7763 0 TD
(.)Tj
/F16 1 Tf
1.5837 0 TD
(If)Tj
/F8 1 Tf
1.1139 0 TD
(u)Tj
/F7 1 Tf
0.9661 0 TD
()Tj
/F2 1 Tf
1.1714 0 TD
(2)Tj
/F16 1 Tf
0.9217 0 TD
(and)Tj
/F8 1 Tf
2.0061 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F16 1 Tf
0.8106 0 TD
[(maintains)-421.6(the)-421.8(same)-421.6(sign)-421.6(thr)51.1(oughout)]TJ
/F2 1 Tf
-18.1153 -1.2694 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
[(u,)-166.6(u)]TJ
/F2 1 Tf
1.8115 0 TD
0.2222 Tc
[(+2)222.2(])]TJ
/F16 1 Tf
2.1356 0 TD
0 Tc
(then)Tj
/F8 1 Tf
2.2233 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
0.2778 Tc
(\)=)Tj
/F8 1 Tf
1.7222 0 TD
0 Tc
(O)Tj
/F2 1 Tf
0.7906 0 TD
(\(1)Tj
/F8 1 Tf
0.8889 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 263.6862 625.4715 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 268.2511 627.1074 Tm
[(\(1\))-166.7(log)]TJ
/F8 1 Tf
2.9027 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\)\))Tj
/F16 1 Tf
1.1356 0 TD
[(for)-357.7(al)-51.1(l)]TJ
/F8 1 Tf
3.0283 0 TD
(x)Tj
/F7 1 Tf
0.8492 0 TD
()Tj
/F8 1 Tf
1.0556 0 TD
(u)Tj
/F2 1 Tf
0.7946 0 TD
0.2222 Tc
(+1)Tj
/F16 1 Tf
1.5 0 TD
0 Tc
(.)Tj
-23.4658 -2.1827 TD
[(Pr)51.2(o)51.1(o)0(f)]TJ
/F2 1 Tf
2.3267 0 TD
[(.)-1122.9(Supp)-27.8(ose)-372.3(not.)-561.4(Let)]TJ
/F8 1 Tf
9.4763 0 TD
(w)Tj
/F2 1 Tf
1.1152 0 TD
[(denote)-372.3(the)-372.3(inm)27.8(um)-372.4(of)-372.3(the)-372.3(set)-372.3(of)-372.3(all)-372.4(sp)-27.7(ecial)]TJ
-14.9272 -1.2694 TD
[(p)-27.8(oin)27.9(ts)-306(at)-306(least)-306(as)-306(large)-305.9(as)]TJ
/F8 1 Tf
11.3054 0 TD
(u)Tj
/F2 1 Tf
0.74 0 TD
[(+)-167.5(1)0(.)-435.3(B)0(y)-306(Lemma)-305.9(3.10)-306(w)27.8(e)-306(kno)27.8(w)-306(that)]TJ
/F8 1 Tf
15.4943 0 TD
(w)Tj
/F7 1 Tf
1.0205 0 TD
()Tj
/F8 1 Tf
1.0556 0 TD
(u)Tj
/F2 1 Tf
0.74 0 TD
[(+)-167.5(2)0(.)-435.3(Let)]TJ
/F8 1 Tf
-30.3559 -1.2694 TD
(v)Tj
/F2 1 Tf
0.8226 0 TD
[(denote)-302(the)-302(inm)27.8(um)-302.1(of)-302.1(p)-27.7(oin)27.8(ts)]TJ
/F8 1 Tf
12.8492 0 TD
(x)Tj
/F7 1 Tf
0.8493 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(u)Tj
/F2 1 Tf
0.7321 0 TD
[(+)-159.7(2)-302.1(suc)27.8(h)-302.1(that)]TJ
/F8 1 Tf
6.0993 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
[(\))-302.1(has)-302.1(the)-302(opp)-27.8(osite)-302.1(sign)]TJ
-24.1643 -1.2694 TD
(to)Tj
/F8 1 Tf
1.1961 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
[(\);)-315.9(if)-307.3(no)-307.2(suc)27.8(h)-307.2(p)-27.7(oin)27.8(t)-307.3(exists)-307.1(set)]TJ
/F8 1 Tf
12.3203 0 TD
(v)Tj
/F2 1 Tf
0.7984 0 TD
(=)Tj
/F7 1 Tf
1.0555 0 TD
()Tj
/F2 1 Tf
1 0 TD
0.4357 Tc
[(.N)435.7(o)463.5(w)]TJ
/F8 1 Tf
2.9652 0 TD
0 Tc
(E)Tj
/F2 1 Tf
1.103 0 TD
[(main)27.8(tains)-307.1(the)-307.2(same)-307.2(sign)-307.2(in)]TJ
-22.366 -1.2694 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
(w)Tj
/F7 1 Tf
0.965 0 TD
()Tj
/F2 1 Tf
1 0 TD
(1)Tj
/F8 1 Tf
0.5 0 TD
0.1667 Tc
(,v)Tj
/F2 1 Tf
0.965 0 TD
0 Tc
[(])-333.4(\(since)-333.3(it)-333.3(is)-333.3(a)-333.3(subin)27.9(terv)55.6(al)-333.3(of)-333.2([)]TJ
/F8 1 Tf
12.825 0 TD
[(u,)-166.6(v)]TJ
/F2 1 Tf
1.5375 0 TD
[(]\))-333.4(and)-333.2(so)]TJ
/F9 1 Tf
-17.3477 -1.0179 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F8 1 Tf
0.3333 0.05 TD
0.0269 Tc
(wE)Tj
/F2 1 Tf
1.5387 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
0.2222 Tc
(\)+)Tj
/F9 1 Tf
1.6111 1.3611 TD
0 Tc
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 187.9362 535.9482 Tm
(v)Tj
-0.6083 -2.7084 TD
(w)Tj
/F8 1 Tf
10.9091 0 0 10.9091 194.662 524.254 Tm
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(t)Tj
/F2 1 Tf
0.3611 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(dt)Tj
/F2 1 Tf
1.1037 0 TD
(+)Tj
1.8433 0.6765 TD
(1)Tj
ET
239.97 527.206 21.463 -0.436 re
f
BT
10.9091 0 0 10.9091 239.9143 516.7711 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.7853 1.636 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 264.7064 511.1074 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 283.1621 531.6952 Tm
(1)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F8 1 Tf
1 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 307.2907 530.0583 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 311.8556 531.6952 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
283.17 527.206 42.667 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 301.7514 516.7711 Tm
(p)Tj
/F2 1 Tf
2.4834 0.6859 TD
(log)Tj
/F8 1 Tf
1.4583 0 TD
0.3334 Tc
(pE)Tj
/F9 1 Tf
1.6323 1.11 TD
0 Tc
( )Tj
/F8 1 Tf
0.5973 -1.11 TD
(w)Tj
/F7 1 Tf
0.965 0 TD
()Tj
/F2 1 Tf
1.1125 0.6765 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(p)Tj
ET
391.65 527.206 21.463 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 391.7061 516.7711 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.6185 1.796 TD
(\012)Tj
0.5973 0.04 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F7 1 Tf
0.6111 0.05 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(w)Tj
/F7 1 Tf
0.7428 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
(|)Tj
/F2 1 Tf
0.2777 0 TD
(;)Tj
-32.9721 -2.6276 TD
[(on)-333.3(the)-333.4(other)-333.3(hand)]TJ
/F9 1 Tf
4.0423 -0.7679 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F2 1 Tf
1.1766 0.7266 TD
(1)Tj
ET
167.73 477.646 21.463 -0.436 re
f
BT
10.9091 0 0 10.9091 167.729 467.1841 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.7853 1.6359 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 192.5211 461.5193 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 210.9758 482.1071 Tm
(1)Tj
/F7 1 Tf
0.7223 0 TD
()Tj
/F8 1 Tf
1 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 235.1053 480.4713 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 239.6702 482.1071 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\))Tj
ET
210.93 477.646 42.667 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 229.5651 467.1841 Tm
(p)Tj
/F2 1 Tf
2.4835 0.6859 TD
(log)Tj
/F8 1 Tf
1.4583 0 TD
0.3334 Tc
(pE)Tj
/F9 1 Tf
1.6323 1.11 TD
0 Tc
( )Tj
/F8 1 Tf
0.5972 -1.11 TD
(v)Tj
/F7 1 Tf
0.7428 0 TD
()Tj
/F2 1 Tf
1.1126 0.6765 TD
(log)Tj
/F8 1 Tf
1.4583 0 TD
(p)Tj
ET
1 i
317.13 477.21 21.463 0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 317.0959 467.1841 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F9 1 Tf
0.6186 1.796 TD
(\012)Tj
0.5973 0.0399 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F7 1 Tf
0.6111 0.05 TD
0.2777 Tc
(|)Tj
/F8 1 Tf
1.3333 0 TD
0 Tc
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F2 1 Tf
0.2778 0 TD
[(\(1)-222.1(+)]TJ
/F8 1 Tf
2.111 0 TD
(o)Tj
/F2 1 Tf
0.4848 0 TD
(\(1\)\))Tj
/F8 1 Tf
1.6667 0 TD
(,)Tj
/F2 1 Tf
-29.6525 -2.6831 TD
(since)Tj
/F7 1 Tf
2.5237 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(v)Tj
/F7 1 Tf
0.792 0 TD
()Tj
/F2 1 Tf
1.049 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(p/)Tj
/F2 1 Tf
1.1698 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
0.4006 Tc
(|)Tj
/F2 1 Tf
1.8567 0 TD
0 Tc
(max)Tj
/F14 1 Tf
7.9701 0 0 7.9701 277.3021 443.7599 Tm
(x)Tj
/F15 1 Tf
0.5981 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(v)Tj
/F15 1 Tf
0.5532 0 TD
()Tj
/F1 1 Tf
0.8265 0 TD
(1)Tj
/F7 1 Tf
10.9091 0 0 10.9091 306.2027 445.3968 Tm
(|)Tj
/F8 1 Tf
0.2777 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
0.4006 Tc
(||)Tj
/F8 1 Tf
2.1344 0 TD
0 Tc
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F2 1 Tf
0.2778 0 TD
[(.)-665.5(Therefore,)-425.3(b)27.8(y)-407(\(3.9\),)]TJ
-24.3302 -1.2694 TD
[(w)27.8(e)-410.8(deduce)-410.8(that)-410.8(\()]TJ
/F8 1 Tf
7.5936 0 TD
(w)Tj
/F7 1 Tf
1.0167 0 TD
()Tj
/F2 1 Tf
1.0516 0 TD
0.2739 Tc
(1+)Tj
/F8 1 Tf
1.8256 0 TD
0 Tc
(o)Tj
/F2 1 Tf
0.4847 0 TD
(\(1\)\))Tj
/F7 1 Tf
1.6667 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\))Tj
/F7 1 Tf
0.389 0 TD
0.4069 Tc
(|)Tj
/F8 1 Tf
1.8694 0 TD
0 Tc
(O)Tj
/F2 1 Tf
0.7905 0 TD
(\(1)Tj
/F8 1 Tf
0.8889 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 352.8767 429.9119 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 357.4416 431.5487 Tm
[(\(1\))-166.7(log)]TJ
/F8 1 Tf
2.9028 0 TD
(z)Tj
/F2 1 Tf
0.5089 0 TD
[(\)\))-410.9(whic)27.8(h)-410.7(pro)27.8(v)27.8(e)0(s)-410.9(the)]TJ
-25.2873 -1.2694 TD
(lemma.)Tj
ET
482.61 424.449 6.775 -0.399 re
f
482.61 424.028 0.399 -5.978 re
f
488.97 424.028 0.399 -5.978 re
f
482.61 418.089 6.775 -0.398 re
f
BT
/F5 1 Tf
10.9091 0 0 10.9091 140.7173 393.8899 Tm
[(Pr)22.3(oposition)-419.9(3.12)]TJ
/F2 1 Tf
8.4505 0 TD
(.)Tj
/F16 1 Tf
1.4082 0 TD
(Fix)Tj
/F8 1 Tf
1.8181 0 TD
0.3449 Tc
(>)Tj
/F2 1 Tf
1.9339 0 TD
0 Tc
(0)Tj
/F16 1 Tf
0.5 0 TD
[(.)-571.1(Given)-394.7(a)-394.8(s)0(p)51.2(e)51(cial)-394.8(p)51.2(oint)]TJ
/F8 1 Tf
10.5138 0 TD
0.3718 Tc
(w>)Tj
/F2 1 Tf
2.2103 0 TD
0.249 Tc
(4+)Tj
/F8 1 Tf
1.7759 0 TD
0 Tc
()Tj
/F16 1 Tf
0.8611 0 TD
(either)Tj
-31.4809 -1.2694 TD
[(ther)51.2(e)-316.6(exists)-316.6(a)-316.7(p)51.1(oint)]TJ
/F8 1 Tf
8.4642 0 TD
()Tj
/F7 1 Tf
0.7613 0 TD
()Tj
/F2 1 Tf
0.9445 0 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
(w)Tj
/F7 1 Tf
0.8755 0 TD
()Tj
/F2 1 Tf
0.9106 0 TD
(1)Tj
/F8 1 Tf
0.5 0 TD
0.1666 Tc
(,w)Tj
/F2 1 Tf
1.1872 0 TD
0 Tc
(])Tj
/F16 1 Tf
0.5944 0 TD
[(for)-316.6(which)]TJ
/F8 1 Tf
4.326 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F16 1 Tf
0.7055 0 TD
(and)Tj
/F8 1 Tf
1.9011 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\))Tj
/F16 1 Tf
0.7055 0 TD
[(have)-316.6(opp)51.2(osite)-316.6(signs)]TJ
-25.7494 -1.2694 TD
(and)Tj
/F7 1 Tf
1.9422 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4835 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
0.2778 Tc
(|)Tj
/F2 1 Tf
1.6111 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F7 1 Tf
0.9651 0 TD
()Tj
/F2 1 Tf
1 0 TD
(4)Tj
/F7 1 Tf
0.7222 0 TD
()Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
0.4663 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F2 1 Tf
0.2778 0 TD
(,)Tj
/F16 1 Tf
0.6355 0 TD
(or)Tj
/F8 1 Tf
1.2905 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
0.2778 Tc
(\)=)Tj
/F8 1 Tf
1.7222 0 TD
0 Tc
(O)Tj
/F2 1 Tf
0.7906 0 TD
(\(1)Tj
/F8 1 Tf
0.8889 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 365.5851 364.5569 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 370.15 366.1938 Tm
[(\(1\))-166.7(log)]TJ
/F8 1 Tf
2.9027 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\)\))Tj
/F16 1 Tf
0.7778 0 TD
(.)Tj
-25.2209 -2.1826 TD
[(Pr)51.2(o)51.1(o)0(f)]TJ
/F2 1 Tf
2.3267 0 TD
[(.)-1572.8(Select)]TJ
/F8 1 Tf
4.8535 0 TD
(v)Tj
/F7 1 Tf
0.9884 0 TD
()Tj
/F2 1 Tf
1.1344 0 TD
([)Tj
/F8 1 Tf
0.2777 0 TD
(w)Tj
/F2 1 Tf
1.0411 0 TD
0.2982 Tc
(+1)Tj
/F8 1 Tf
1.576 0 TD
0.1666 Tc
(,w)Tj
/F2 1 Tf
1.4855 0 TD
0 Tc
[(+)-298.2(3)0(])-447.3(with)]TJ
/F8 1 Tf
4.6929 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(v)Tj
/F2 1 Tf
0.5205 0 TD
0.4678 Tc
(\)=)Tj
/F8 1 Tf
2.1022 0 TD
0 Tc
(O)Tj
/F2 1 Tf
0.7906 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(z)Tj
/F15 1 Tf
7.9701 0 0 7.9701 401.1407 346.3422 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
(v)Tj
/F2 1 Tf
10.9091 0 0 10.9091 412.6346 342.3831 Tm
[(\))-447.4(\(if)]TJ
/F8 1 Tf
2.2558 0 TD
(v)Tj
/F2 1 Tf
0.9679 0 TD
[(do)-27.7(es)-447.3(not)]TJ
-30.1586 -1.2694 TD
[(exist)-350.2(then)-350.2(the)-350.3(prop)-27.8(osition)-350.1(follo)27.8(ws)-350.3(from)-350.2(Lemma)-350.2(3.11\).)-495.1(Cho)-27.8(ose)]TJ
/F8 1 Tf
27.3945 0 TD
()Tj
/F2 1 Tf
0.7882 0 TD
(=)Tj
/F7 1 Tf
1.0838 0 TD
()Tj
/F2 1 Tf
0.7777 0 TD
[(1)-350.3(s)0(o)-350.2(that)]TJ
/F8 1 Tf
-30.0443 -1.2694 TD
0.0378 Tc
(E)Tj
/F2 1 Tf
1.2781 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\))Tj
/F8 1 Tf
0.6667 0 TD
(>)Tj
/F2 1 Tf
1.0555 0 TD
[(0.)-417.7(Supp)-27.8(ose)-253(that)]TJ
/F8 1 Tf
7.1237 0 TD
0.0378 Tc
(E)Tj
/F2 1 Tf
1.2781 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F8 1 Tf
0.6667 0 TD
(>)Tj
/F7 1 Tf
1.0555 0 TD
()Tj
/F2 1 Tf
0.7778 0 TD
(\(2)Tj
/F8 1 Tf
0.8889 0 TD
(w)Tj
/F7 1 Tf
0.8044 0 TD
()Tj
/F8 1 Tf
0.8393 0 TD
(v)Tj
/F7 1 Tf
0.5821 0 TD
()Tj
/F2 1 Tf
0.8393 0 TD
(1)Tj
/F7 1 Tf
0.5616 0 TD
()Tj
/F8 1 Tf
0.8393 0 TD
()Tj
/F2 1 Tf
0.4663 0 TD
(\))Tj
/F8 1 Tf
0.389 0 TD
0.0378 Tc
(E)Tj
/F2 1 Tf
1.2781 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.389 0 TD
(w)Tj
/F2 1 Tf
0.7427 0 TD
[(\))-253(for)-253(all)]TJ
/F8 1 Tf
3.4006 0 TD
(x)Tj
/F7 1 Tf
0.8493 0 TD
()Tj
/F2 1 Tf
0.9445 0 TD
([)Tj
/F8 1 Tf
0.2777 0 TD
(w)Tj
/F7 1 Tf
0.8044 0 TD
()Tj
/F2 1 Tf
0.8393 0 TD
(1)Tj
/F8 1 Tf
0.5 0 TD
0.1667 Tc
(,w)Tj
/F2 1 Tf
1.1873 0 TD
-0.0001 Tc
(].)Tj
-33.4171 -1.2694 TD
0 Tc
(Then)Tj
/F8 1 Tf
2.6111 0 TD
()Tj
/F2 1 Tf
0.8156 0 TD
[(times)-333.3(the)-333.3(righ)27.8(t)-333.3(side)-333.3(of)-333.3(\(3.9\))-333.3(is)]TJ
/F7 1 Tf
-2.3723 -2.1224 TD
()Tj
/F8 1 Tf
1.0555 0 TD
0.0378 Tc
(E)Tj
/F2 1 Tf
1.2782 0 TD
0 Tc
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\)\()Tj
/F8 1 Tf
0.7778 0 TD
(w)Tj
/F7 1 Tf
0.965 0 TD
()Tj
/F9 1 Tf
1 1.3611 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 208.9375 289.3798 Tm
(v)Tj
-0.6084 -2.7084 TD
(w)Tj
/F2 1 Tf
10.9091 0 0 10.9091 215.6633 277.6857 Tm
(1)Tj
/F8 1 Tf
0.5 0 TD
(dt)Tj
/F7 1 Tf
1.1038 0 TD
()Tj
/F2 1 Tf
1.8432 0.6765 TD
(1)Tj
ET
245.25 280.606 21.463 -0.436 re
f
BT
10.9091 0 0 10.9091 245.2643 270.2027 Tm
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F9 1 Tf
0.7853 1.6359 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 270.0563 264.538 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F2 1 Tf
10.9091 0 0 10.9091 288.5111 285.066 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(p)Tj
ET
288.57 280.606 21.398 -0.436 re
f
BT
10.9091 0 0 10.9091 296.4653 270.2027 Tm
(p)Tj
/F2 1 Tf
1.6753 0.6859 TD
(\(2)Tj
/F8 1 Tf
0.8889 0 TD
(w)Tj
/F7 1 Tf
0.965 0 TD
()Tj
/F8 1 Tf
1 0 TD
(v)Tj
/F7 1 Tf
0.7428 0 TD
()Tj
/F2 1 Tf
1 0 TD
(1)Tj
/F7 1 Tf
0.7223 0 TD
()Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
0.6885 0 TD
[(+)-222.2(1\)\))]TJ
/F7 1 Tf
2.5555 0 TD
()Tj
/F8 1 Tf
1.0556 0 TD
()Tj
/F7 1 Tf
0.4663 0 TD
(|)Tj
/F8 1 Tf
0.2777 0 TD
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
(2)Tj
-32.4182 -2.6831 TD
[(since)-279.4(eac)27.8(h)]TJ
/F8 1 Tf
4.5921 0 TD
(v)Tj
/F7 1 Tf
0.635 0 TD
()Tj
/F2 1 Tf
0.8921 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(p/)Tj
/F2 1 Tf
1.1698 0 TD
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F7 1 Tf
0.7868 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(w)Tj
/F2 1 Tf
1.0223 0 TD
[(so)-279.3(that)]TJ
/F7 1 Tf
3.2866 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(E)Tj
/F2 1 Tf
0.7958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(v)Tj
/F7 1 Tf
0.635 0 TD
()Tj
/F2 1 Tf
0.8921 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(p/)Tj
/F2 1 Tf
1.1698 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
(z)Tj
/F2 1 Tf
0.509 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
0.2777 Tc
(||)Tj
/F8 1 Tf
1.8889 0 TD
0 Tc
(E)Tj
/F2 1 Tf
0.7959 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(w)Tj
/F2 1 Tf
0.7428 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
(|)Tj
/F2 1 Tf
0.2778 0 TD
[(.)-426.4(The)-279.4(result)]TJ
-28.814 -1.2694 TD
[(follo)27.8(ws)-333.4(from)-333.2(\(3.9\))-333.3(since)-333.2(2)]TJ
/F8 1 Tf
10.9861 0 TD
(w)Tj
/F7 1 Tf
0.965 0 TD
()Tj
/F8 1 Tf
1 0 TD
(v)Tj
/F7 1 Tf
0.7428 0 TD
()Tj
/F2 1 Tf
1 0 TD
(1)Tj
/F7 1 Tf
0.7778 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(w)Tj
/F7 1 Tf
0.965 0 TD
()Tj
/F2 1 Tf
1 0 TD
(4.)Tj
ET
482.61 241.329 6.775 -0.399 re
f
482.61 240.908 0.399 -5.978 re
f
488.97 240.908 0.399 -5.978 re
f
482.61 234.969 6.775 -0.399 re
f
BT
10.9091 0 0 10.9091 140.7173 212.5031 Tm
[(Giv)27.8(en)]TJ
/F8 1 Tf
2.9012 0 TD
0.2865 Tc
(u>)Tj
/F2 1 Tf
1.9234 0 TD
0.2258 Tc
(3+)Tj
/F8 1 Tf
1.7293 0 TD
0 Tc
()Tj
/F2 1 Tf
0.8049 0 TD
[(w)27.8(e)-338.7(can)-338.6(tak)27.7(e)]TJ
/F8 1 Tf
5.4881 0 TD
(w)Tj
/F2 1 Tf
1.0815 0 TD
[(to)-338.7(b)-27.7(e)-338.7(a)-338.6(s)0(p)-27.8(ecial)-338.5(p)-27.8(oin)27.8(t)-338.6(i)0(n)-338.6([)]TJ
/F8 1 Tf
10.7596 0 TD
(u)Tj
/F2 1 Tf
0.7982 0 TD
0.2258 Tc
(+1)Tj
/F8 1 Tf
1.5036 0 TD
0.1667 Tc
(,u)Tj
/F2 1 Tf
1.2426 0 TD
0 Tc
[(+)-225.8(2)0(])-338.6(and)]TJ
-30.2416 -1.2694 TD
(then)Tj
/F8 1 Tf
2.2778 0 TD
()Tj
/F7 1 Tf
0.7613 0 TD
()Tj
/F2 1 Tf
0.9444 0 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
[(u,)-166.6(u)]TJ
/F2 1 Tf
1.8116 0 TD
0.2222 Tc
[(+2)222.2(])222.3(.)]TJ
/F6 1 Tf
2.4961 -2.9132 TD
0 Tc
[(4.)-574.9(Pro)-31.9(of)-383.3(of)-383.4(Theorems)-383.2(2.4)-383.3(and)-383.3(2.5)]TJ
/F2 1 Tf
-6.5598 -2.3653 TD
[(W)83.3(e)-430.8(shall)-430.7(only)-430.7(pro)27.8(vide)-430.7(the)-430.8(pro)-27.7(of)-430.8(of)-430.9(Theorem)-430.7(2.4,)-455.1(the)-430.8(pro)-27.7(of)-430.8(of)-430.8(Theorem)]TJ
-2.0091 -1.2694 TD
[(2.5)-409.9(b)-27.8(eing)-409.8(en)27.8(tirely)-409.9(similar.)-674(W)83.3(e)-409.8(tak)27.7(e)]TJ
/F8 1 Tf
16.0958 0 TD
(q)Tj
/F2 1 Tf
0.8921 0 TD
[(to)-409.9(b)-27.7(e)-409.9(the)-409.9(pro)-27.7(duct)-409.8(of)-409.9(all)-409.9(the)-409.8(primes)-409.8(in)]TJ
ET
endstream
endobj
101 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F6 9 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F14 22 0 R
/F15 23 0 R
/F16 24 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
103 0 obj
<<
/Length 17237
>>
stream
BT
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
(26)Tj
/F5 1 Tf
7.9701 0 0 7.9701 202.119 726.9739 Tm
[(ANDREW)-377.5(GRANVILLE)-377.4(AND)-377.7(K)0(.)-377.7(SOUND)29.4(ARARAJAN)]TJ
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 701.071 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(z)Tj
/F1 1 Tf
7.9701 0 0 7.9701 128.5948 705.0302 Tm
(1)Tj
/F15 1 Tf
0.5313 0 TD
()Tj
/F14 1 Tf
0.8265 0 TD
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 144.4493 701.071 Tm
0.1667 Tc
(,z)Tj
/F2 1 Tf
0.9534 0 TD
0 Tc
[(\))-342.3(s)0(o)-342.3(that)]TJ
/F8 1 Tf
4.1436 0 TD
(q)Tj
/F7 1 Tf
0.7751 0 TD
()Tj
/F9 1 Tf
1.0704 0.75 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 230.4887 697.8451 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F8 1 Tf
10.9091 0 0 10.9091 247.9303 701.071 Tm
(p)Tj
/F7 1 Tf
0.7958 0 TD
()Tj
/F8 1 Tf
1.0705 0 TD
(x)Tj
/F1 1 Tf
7.9701 0 0 7.9701 274.5245 705.0302 Tm
(1)Tj
/F14 1 Tf
0.5313 0 TD
(/)Tj
/F1 1 Tf
0.5312 0 TD
(3+)Tj
/F14 1 Tf
1.3577 0 TD
(o)Tj
/F1 1 Tf
0.5135 0 TD
(\(1\))Tj
/F2 1 Tf
10.9091 0 0 10.9091 309.2254 701.071 Tm
[(.)-471.4(Since)]TJ
/F7 1 Tf
3.3692 0 TD
0.3677 Tc
(S)Tj
/F2 1 Tf
2.0437 0 TD
0 Tc
([1)Tj
/F8 1 Tf
0.7777 0 TD
(,)Tj
/F2 1 Tf
0.4445 0 TD
[(log)-180.6(log)]TJ
/F8 1 Tf
2.9166 0 TD
(x)Tj
/F2 1 Tf
0.5716 0 TD
[(])-342.2(w)27.7(e)-342.2(also)-342.3(kno)27.8(w)]TJ
-27.5791 -1.2694 TD
[(that)-280.8(\()]TJ
/F8 1 Tf
2.5031 0 TD
0.0359 Tc
(q,)Tj
/F7 1 Tf
0.9267 0 TD
0 Tc
(S)Tj
/F2 1 Tf
0.6805 0 TD
[(\))-277.8(=)-277.8(1)0(.)-426.9(Since)]TJ
/F8 1 Tf
5.4856 0 TD
(q)Tj
/F2 1 Tf
0.7631 0 TD
[(is)-280.8(squarefree)-280.8(w)27.8(e)-280.8(c)27.7(hec)27.9(k)-280.9(that)]TJ
/F8 1 Tf
11.8098 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 365.9816 685.587 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 370.5464 687.2229 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(n)Tj
/F2 1 Tf
0.6003 0 TD
[(\))-280.8(\(as)-280.8(dened)-280.8(in)-280.8(Section)]TJ
-24.0659 -1.2694 TD
[(2\))-382.2(satises)]TJ
/F8 1 Tf
5.0034 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 178.7228 671.739 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 183.2877 673.3749 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 193.0182 677.334 Tm
(k)Tj
/F2 1 Tf
10.9091 0 0 10.9091 198.138 673.3749 Tm
0.3593 Tc
[(\)=1)-22.9(i)359.3(f)]TJ
/F8 1 Tf
3.733 0 TD
0 Tc
(p)Tj
/F17 1 Tf
0.8624 0 TD
()Tj
/F8 1 Tf
0.6371 0 TD
(q)Tj
/F2 1 Tf
0.8646 0 TD
(and)Tj
/F8 1 Tf
1.9933 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 291.737 671.739 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 296.3019 673.3749 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 306.0324 677.334 Tm
(k)Tj
/F2 1 Tf
10.9091 0 0 10.9091 311.1522 673.3749 Tm
0.3592 Tc
(\)=)Tj
/F8 1 Tf
1.8852 0 TD
0 Tc
(h)Tj
/F2 1 Tf
0.5762 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
(\)\()Tj
/F8 1 Tf
0.7778 0 TD
(p)Tj
/F7 1 Tf
0.758 0 TD
()Tj
/F2 1 Tf
1.0325 0 TD
(1\))Tj
/F8 1 Tf
0.8889 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F7 1 Tf
0.758 0 TD
()Tj
/F8 1 Tf
1.0325 0 TD
(h)Tj
/F2 1 Tf
0.5762 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
[(\)\))-382.2(if)]TJ
/F8 1 Tf
2.1256 0 TD
(p)Tj
/F7 1 Tf
0.5031 0 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(q)Tj
/F2 1 Tf
0.8645 0 TD
(and)Tj
/F8 1 Tf
-32.3615 -1.2694 TD
(k)Tj
/F7 1 Tf
0.8299 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
[(1)-289.2(\(note)-289.1(that)]TJ
/F8 1 Tf
5.4786 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 204.4752 657.8909 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 209.0401 659.5268 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(p)Tj
/F14 1 Tf
7.9701 0 0 7.9701 218.7716 663.486 Tm
(k)Tj
/F2 1 Tf
10.9091 0 0 10.9091 223.8914 659.5268 Tm
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(h)Tj
/F2 1 Tf
0.5762 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5032 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
[(1)-289.1(i)0(n)-289.2(this)-289.1(case\))-289.1(.)-429.7(T)0(h)27.8(u)0(s)-289.2(0)]TJ
/F7 1 Tf
10.2533 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 406.1938 657.8909 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 410.7587 659.5268 Tm
(\()Tj
/F8 1 Tf
0.389 0 TD
(n)Tj
/F2 1 Tf
0.6002 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F2 1 Tf
1.0556 0 TD
[(1)-289.1(for)-289.1(all)]TJ
/F8 1 Tf
3.6203 0 TD
(n)Tj
/F2 1 Tf
0.6002 0 TD
(,)Tj
-33.6948 -1.2694 TD
[(and)-365.2(w)27.8(e)-365.2(ma)27.8(y)-365.2(apply)-365.2(the)-365.2(results)-365.1(of)-365.2(Section)-365.2(3,)-373.2(in)-365.2(particular)-365.1(Corollary)-365.2(3.2.)-540.1(F)83.4(rom)]TJ
T*
[(Corollary)-292.9(3.2)-292.9(w)27.7(e)-292.9(obtain)-292.9(that)-292.9(for)]TJ
/F7 1 Tf
14.0718 0.7203 TD
()Tj
ET
281.37 640.126 5.553 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 281.4008 631.8307 Tm
(z)Tj
/F7 1 Tf
0.7868 0 TD
()Tj
/F8 1 Tf
1.0556 0 TD
(u)Tj
/F7 1 Tf
0.8502 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(5)Tj
/F8 1 Tf
0.5 0 TD
(/ )Tj
/F1 1 Tf
7.9701 0 0 7.9701 339.0067 635.7898 Tm
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 346.935 631.8307 Tm
[(there)-292.9(exists)]TJ
/F8 1 Tf
5.2387 0 TD
()Tj
/F7 1 Tf
0.8611 0 TD
()Tj
/F2 1 Tf
0.9445 0 TD
([)Tj
/F8 1 Tf
0.2778 0 TD
[(u,)-166.6(u)]TJ
/F2 1 Tf
1.5893 0 TD
0.1415 Tc
[(\()141.5(1+2)141.5(2)]TJ
/F8 1 Tf
2.9497 0 TD
0 Tc
( )Tj
/F2 1 Tf
0.5324 0 TD
(\)])Tj
-33.3059 -1.2694 TD
[(suc)27.8(h)-333.3(that)]TJ
/F9 1 Tf
4.7105 -0.7877 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F2 1 Tf
0.6966 0.7265 TD
(1)Tj
ET
175.05 599.806 10.991 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 175.0187 589.3579 Tm
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 180.5718 592.5091 Tm
()Tj
/F9 1 Tf
10.9091 0 0 10.9091 191.4869 607.208 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 189.0232 583.2977 Tm
(n)Tj
/F15 1 Tf
0.6448 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
(z)Tj
/F10 1 Tf
4.9813 0 0 4.9813 205.0232 585.5662 Tm
()Tj
/F8 1 Tf
10.9091 0 0 10.9091 211.5258 596.8439 Tm
(f)Tj
/F14 1 Tf
7.9701 0 0 7.9701 216.8668 595.208 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 221.4316 596.8439 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
(n)Tj
/F2 1 Tf
0.6003 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1 0 TD
(G)Tj
/F14 1 Tf
7.9701 0 0 7.9701 258.3751 595.208 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 262.94 596.8439 Tm
(\(1\))Tj
/F9 1 Tf
1.2777 1.15 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F7 1 Tf
0.6111 0.05 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(exp)Tj
/F9 1 Tf
1.6945 1.1101 TD
( )Tj
/F7 1 Tf
0.8194 -1.1101 TD
()Tj
/F8 1 Tf
1 0 TD
(u)Tj
/F2 1 Tf
0.5725 0 TD
[(\(1)-222.1(+)-222.2(2)0(5)]TJ
/F8 1 Tf
3.1111 0 TD
( )Tj
/F2 1 Tf
0.5324 0 TD
[(\))-166.7(log)]TJ
/F9 1 Tf
2.0139 1.11 TD
( )Tj
/F2 1 Tf
0.7068 -0.4335 TD
(2)Tj
/F8 1 Tf
0.5 0 TD
(u)Tj
ET
409.05 599.806 11.699 -0.436 re
f
BT
10.9091 0 0 10.9091 409.6469 589.3609 Tm
( )Tj
/F1 1 Tf
7.9701 0 0 7.9701 415.4551 592.5121 Tm
(2)Tj
/F9 1 Tf
10.9091 0 0 10.9091 421.9616 608.9535 Tm
(\012\012)Tj
/F8 1 Tf
1.1945 -1.1101 TD
(.)Tj
/F2 1 Tf
-28.9844 0 TD
(\(4.1\))Tj
2.0091 -2.952 TD
[(Using)-261.7(Prop)-27.7(osition)-261.7(2.2,)-276(w)27.8(e)-261.8(ma)27.8(y)-261.7(easily)-261.7(nd)]TJ
/F8 1 Tf
17.9911 0 TD
()Tj
/F7 1 Tf
0.6944 0 TD
()Tj
/F2 1 Tf
0.9445 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x/)Tj
/F2 1 Tf
1.0716 0 TD
(\(2\()Tj
/F8 1 Tf
1.2777 0 TD
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 390.2864 568.5997 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 396.5868 564.6406 Tm
[(+)-79.1(1\)\))]TJ
/F8 1 Tf
2.1346 0 TD
0.1667 Tc
[(,x)166.7(/)]TJ
/F2 1 Tf
1.516 0 TD
0 Tc
(\(2)Tj
/F8 1 Tf
0.8889 0 TD
(z)Tj
/F14 1 Tf
7.9701 0 0 7.9701 451.6613 568.5997 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 457.0999 564.6406 Tm
[(\)\))-261.8(suc)27.8(h)]TJ
-31.0109 -1.2694 TD
(that)Tj
/F8 1 Tf
2.1373 0 TD
()Tj
/F2 1 Tf
0.7206 0 TD
[(is)-303.9(not)-303.9(divisible)-303.8(b)27.8(y)-304(an)27.8(y)-303.9(prime)-303.9(b)-27.7(elo)27.7(w)-303.9(\(log)]TJ
/F8 1 Tf
17.2942 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
[(3.)-434.6(Suc)27.8(h)-303.9(a)0(n)]TJ
/F8 1 Tf
4.9592 0 TD
()Tj
/F2 1 Tf
0.7206 0 TD
[(is)-303.8(coprime)-303.9(to)]TJ
/F8 1 Tf
5.9201 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(,)Tj
-33.6948 -1.2694 TD
[(satises)-362.6(\()]TJ
/F8 1 Tf
4.1015 0 TD
(,)Tj
/F7 1 Tf
0.8611 0 TD
(S)Tj
/F2 1 Tf
0.6805 0 TD
[(\))-326.5(=)-326.4(1,)-369.9(and)-362.6(further,)]TJ
/F8 1 Tf
8.622 0 TD
()Tj
/F2 1 Tf
0.5958 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4167 0 TD
(\))Tj
/F8 1 Tf
0.3889 0 TD
(/)Tj
/F7 1 Tf
1.2432 0 TD
()Tj
/F2 1 Tf
1.3265 0 TD
[(1.)-532.1(Prop)-27.8(osition)-362.5(2.2)-362.6(therefore)-362.5(yields,)]TJ
-18.625 -1.2694 TD
[(b)27.8(y)-333.3(\(4.1\),)]TJ
7.8211 -1.6623 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 213.2115 503.3265 Tm
(q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 220.2003 504.9624 Tm
0.2222 Tc
(+)Tj
/F14 1 Tf
7.9701 0 0 7.9701 240.2002 503.1851 Tm
0 Tc
()Tj
/F7 1 Tf
10.9091 0 0 10.9091 247.188 504.9624 Tm
()Tj
/F2 1 Tf
1.2778 0 TD
(exp)Tj
/F9 1 Tf
1.6944 1.11 TD
( )Tj
/F7 1 Tf
0.8195 -1.11 TD
()Tj
/F8 1 Tf
1 0 TD
(u)Tj
/F2 1 Tf
0.5724 0 TD
0.2222 Tc
[(\()222.2(1+2)222.2(5)]TJ
/F8 1 Tf
3.1111 0 TD
0 Tc
( )Tj
/F2 1 Tf
0.5323 0 TD
[(\))-166.7(log)]TJ
/F9 1 Tf
2.014 1.11 TD
( )Tj
/F2 1 Tf
0.7068 -0.4335 TD
(2)Tj
/F8 1 Tf
0.5 0 TD
(u)Tj
ET
375.09 507.886 11.699 -0.436 re
f
BT
10.9091 0 0 10.9091 375.7131 497.4795 Tm
( )Tj
/F1 1 Tf
7.9701 0 0 7.9701 381.5213 500.6306 Tm
(2)Tj
/F9 1 Tf
10.9091 0 0 10.9091 388.0289 517.072 Tm
(\012\012)Tj
/F8 1 Tf
1.1944 -1.11 TD
(,)Tj
/F2 1 Tf
-25.8738 -2.206 TD
[(pro)27.8(ving)-333.3(Theorem)-333.3(2.4)-333.3(\(in)-333.3(the)-333.3(statemen)27.8(t)-333.3(o)0(f)-333.3(whic)27.8(h)-333.3(w)27.8(e)-333.4(migh)27.8(t)-333.3(tak)27.8(e)]TJ
/F8 1 Tf
27.8722 0 TD
()Tj
/F2 1 Tf
0.6944 0 TD
(=)Tj
/F8 1 Tf
1.0556 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\).)Tj
/F6 1 Tf
-25.1437 -2.9132 TD
[(5.)-574.9(Limitations)-383.2(to)-383.3(the)-383.3(equidistribution)-383.1(of)-383.3(primes)]TJ
/F2 1 Tf
-2.9517 -2.3653 TD
[(In)-409.3(this)-409.1(section)-409.2(w)27.8(e)-409.3(will)-409.2(exploit)-409.2(the)-409.2(exibilit)27.9(y)-409.2(aorded)-409.2(b)27.7(y)-409.2(our)-409.2(general)-409.1(os-)]TJ
-2.0091 -1.2694 TD
[(cillation)-326.2(results)-326.2(to)-326.3(obtain)-326.3(renemen)27.8(ts)-326.2(to)-326.3(previous)-326.2(results)-326.2(on)-326.3(the)-326.2(limitations)-326.2(to)]TJ
T*
[(the)-386.2(equidistribution)-385.9(of)-386.1(primes.)-602.7(W)83.3(e)-386.2(stated)-386.1(in)-386.1(Theorem)-386.1(1.5)-386.1(\(see)-386.1(In)27.7(tro)-27.7(duction\))]TJ
T*
[(the)-320.1(result)-320(for)-320.1(primes)-320(in)-320.2(short)-320.1(in)27.9(terv)55.6(als)-320.1(and)-320.1(w)27.8(e)-320.1(no)27.8(w)-320.2(state)-320.1(the)-320(analogous)-320(results)]TJ
T*
[(for)-333.3(primes)-333.2(in)-333.3(arithmetic)-333.3(progressions.)]TJ
/F5 1 Tf
2.0091 -2.1826 TD
[(Theorem)-302.7(5.1)]TJ
/F2 1 Tf
6.2271 0 TD
(.)Tj
/F16 1 Tf
1.1191 0 TD
-0.0511 Tc
[(Le)-51.1(t)]TJ
/F8 1 Tf
1.6604 0 TD
0 Tc
()Tj
/F16 1 Tf
0.7087 0 TD
[(b)51(e)-292(lar)51.1(ge)-292(and)-292(supp)51.1(ose)-291.9(that)]TJ
/F8 1 Tf
10.9541 0 TD
()Tj
/F16 1 Tf
0.7088 0 TD
[(has)-292(fewer)-292(than)]TJ
/F2 1 Tf
6.5366 0 TD
(\(log)Tj
/F8 1 Tf
1.8472 0 TD
()Tj
/F2 1 Tf
0.4166 0 TD
(\))Tj
/F1 1 Tf
7.9701 0 0 7.9701 474.1808 348.0687 Tm
(1)Tj
/F15 1 Tf
0.5312 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
()Tj
/F16 1 Tf
10.9091 0 0 10.9091 118.7995 330.2615 Tm
[(prime)-357.7(divisors)-357.7(b)51.1(elow)]TJ
/F2 1 Tf
9.2255 0 TD
(log)Tj
/F8 1 Tf
1.4584 0 TD
()Tj
/F16 1 Tf
0.4166 0 TD
[(.)-460(Supp)51.1(ose)-357.7(that)]TJ
/F2 1 Tf
-3.3845 -1.7948 TD
(\(log)Tj
/F8 1 Tf
1.8473 0 TD
()Tj
/F2 1 Tf
0.4166 0 TD
(\))Tj
/F1 1 Tf
7.9701 0 0 7.9701 231.9133 315.186 Tm
(1+)Tj
/F14 1 Tf
1.3577 0 TD
()Tj
/F7 1 Tf
10.9091 0 0 10.9091 250.1729 310.6819 Tm
()Tj
/F8 1 Tf
1.0556 0 TD
(y)Tj
/F7 1 Tf
0.8039 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(exp\()Tj
/F8 1 Tf
1.9167 0 TD
(\012)Tj
/F9 1 Tf
0.6184 0.9038 TD
()Tj
ET
1 i
320.49 320.49 20.454 0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 320.538 310.6819 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(/)Tj
/F9 1 Tf
0.9166 0.9038 TD
()Tj
ET
357.33 320.926 43.636 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 357.3559 310.6819 Tm
[(2)-166.7(log)-180.6(log)]TJ
/F8 1 Tf
3.5834 0 TD
()Tj
/F2 1 Tf
0.4166 0 TD
(\))Tj
/F16 1 Tf
-25.8677 -1.7949 TD
[(for)-401.6(a)-401.7(c)51.1(ertain)-401.6(absolute)-401.5(c)51.1(onstant)]TJ
/F8 1 Tf
13.8659 0 TD
0.4101 Tc
(\012>)Tj
/F2 1 Tf
2.1107 0 TD
0 Tc
(0,)Tj
/F16 1 Tf
1.1904 0 TD
[(and)-401.6(put)]TJ
/F8 1 Tf
3.7678 0 TD
(x)Tj
/F2 1 Tf
0.9288 0 TD
(=)Tj
/F8 1 Tf
1.135 0 TD
(y)Tj
/F16 1 Tf
0.9428 0 TD
[(.)-591.6(Dene)-401.6(for)-401.6(inte)51.2(gers)]TJ
/F8 1 Tf
9.5027 0 TD
(a)Tj
/F16 1 Tf
-33.444 -1.2694 TD
[(c)51.1(oprime)-357.7(to)]TJ
/F8 1 Tf
4.9961 0 TD
()Tj
/F2 1 Tf
3.2683 -1.7357 TD
(\()Tj
/F8 1 Tf
1.2222 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(;)Tj
/F8 1 Tf
0.4445 0 TD
[(,)-166.6(a)]TJ
/F2 1 Tf
1.3897 0 TD
0.2778 Tc
(\)=)Tj
/F9 1 Tf
1.7222 1.4099 TD
0 Tc
()Tj
/F8 1 Tf
0.7362 -1.4099 TD
()Tj
/F2 1 Tf
0.5914 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5716 0 TD
(;)Tj
/F8 1 Tf
0.4444 0 TD
[(,)-166.6(a)]TJ
/F2 1 Tf
1.3897 0 TD
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1.7481 0.6764 TD
(x)Tj
ET
331.05 260.85 20.166 0.436 re
f
BT
10.9091 0 0 10.9091 331.0605 250.8344 Tm
(\015)Tj
/F2 1 Tf
0.6542 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
()Tj
/F2 1 Tf
0.4166 0 TD
(\))Tj
/F9 1 Tf
0.4984 2.096 TD
0.1667 Tc
()Tj
/F8 1 Tf
2.6954 -0.7335 TD
0 Tc
(x)Tj
ET
374.85 261.286 20.166 -0.436 re
f
BT
10.9091 0 0 10.9091 374.8603 250.8344 Tm
(\015)Tj
/F2 1 Tf
0.6542 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
()Tj
/F2 1 Tf
0.4166 0 TD
(\))Tj
/F8 1 Tf
0.4985 0.686 TD
(.)Tj
/F16 1 Tf
-25.4305 -2.4365 TD
[(Ther)51.2(e)-397(exist)-396.9(numb)51.1(ers)]TJ
/F8 1 Tf
9.2957 0 TD
(x)Tj
/F15 1 Tf
7.9701 0 0 7.9701 226.4428 230.1011 Tm
()Tj
/F16 1 Tf
10.9091 0 0 10.9091 237.8571 231.738 Tm
[(in)-396.9(the)-396.9(interval)]TJ
/F2 1 Tf
6.6724 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
[(x,)-166.7(xy)]TJ
/F14 1 Tf
7.9701 0 0 7.9701 337.9477 235.6962 Tm
0.0296 Tc
(D/)Tj
/F1 1 Tf
1.6096 0 TD
0 Tc
[(log)-14.7(\(log)]TJ
/F14 1 Tf
3.3351 0 TD
0.0358 Tc
(y/)Tj
/F1 1 Tf
1.2694 0 TD
0 Tc
[(log)-191.8(log)]TJ
/F14 1 Tf
3.099 0 TD
()Tj
/F1 1 Tf
0.434 0 TD
(\))Tj
/F2 1 Tf
10.9091 0 0 10.9091 419.4252 231.738 Tm
(\),)Tj
/F16 1 Tf
1.0733 0 TD
[(and)-396.9(inte)51.1(gers)]TJ
/F8 1 Tf
-28.6307 -1.2694 TD
(a)Tj
/F15 1 Tf
7.9701 0 0 7.9701 124.5659 216.2531 Tm
()Tj
/F16 1 Tf
10.9091 0 0 10.9091 135.5537 217.8899 Tm
[(c)51.1(oprime)-357.7(to)]TJ
/F8 1 Tf
4.9961 0 TD
()Tj
/F16 1 Tf
0.7744 0 TD
[(such)-357.7(that)]TJ
/F2 1 Tf
-3.6485 -1.7949 TD
(\()Tj
/F8 1 Tf
1.2223 0 TD
(x)Tj
/F1 1 Tf
7.9701 0 0 7.9701 178.2715 196.6734 Tm
(+)Tj
/F2 1 Tf
10.9091 0 0 10.9091 185.3559 198.3094 Tm
(;)Tj
/F8 1 Tf
0.4445 0 TD
[(,)-166.6(a)]TJ
/F1 1 Tf
7.9701 0 0 7.9701 205.3649 196.6734 Tm
(+)Tj
/F2 1 Tf
10.9091 0 0 10.9091 212.4493 198.3094 Tm
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(y)Tj
/F15 1 Tf
7.9701 0 0 7.9701 236.9773 202.8144 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
()Tj
/F1 1 Tf
0.5076 0 TD
(\()Tj
/F14 1 Tf
0.4132 0 TD
(,y)Tj
/F1 1 Tf
1.2902 0 TD
(\))Tj
/F8 1 Tf
10.9091 0 0 10.9091 264.9774 198.3094 Tm
(,)Tj
/F16 1 Tf
2.4889 0 TD
(and)Tj
/F2 1 Tf
3.629 0 TD
(\()Tj
/F8 1 Tf
1.2222 0 TD
(x)Tj
/F15 1 Tf
7.9701 0 0 7.9701 351.2847 196.6734 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 358.3691 198.3094 Tm
(;)Tj
/F8 1 Tf
0.4445 0 TD
[(,)-166.6(a)]TJ
/F15 1 Tf
7.9701 0 0 7.9701 378.3781 196.6734 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 385.4625 198.3094 Tm
(\))Tj
/F7 1 Tf
0.6667 0 TD
0.2778 Tc
()Tj
/F8 1 Tf
1.8333 0 TD
0 Tc
(y)Tj
/F15 1 Tf
7.9701 0 0 7.9701 418.4757 202.8144 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
()Tj
/F1 1 Tf
0.5076 0 TD
(\()Tj
/F14 1 Tf
0.4132 0 TD
(,y)Tj
/F1 1 Tf
1.2903 0 TD
(\))Tj
/F8 1 Tf
10.9091 0 0 10.9091 446.4757 198.3094 Tm
(.)Tj
/F16 1 Tf
-28.0279 -1.7948 TD
[(Her)51.2(e)]TJ
/F8 1 Tf
2.3098 0 TD
(D)Tj
/F16 1 Tf
1.1316 0 TD
[(is)-275.9(an)-275.9(absolute)-275.8(p)51.1(ositive)-275.8(c)51.1(onstant)-275.8(which)-275.8(dep)51.1(ends)-275.8(only)-275.9(on)]TJ
/F8 1 Tf
23.3364 0 TD
()Tj
/F16 1 Tf
0.4663 0 TD
[(,)-292.3(and)]TJ
/F8 1 Tf
2.4593 0 TD
()Tj
/F2 1 Tf
0.4823 0 TD
(\()Tj
/F7 1 Tf
0.389 0 TD
()Tj
/F8 1 Tf
0.2777 0 TD
(,)Tj
/F7 1 Tf
0.4445 0 TD
()Tj
/F2 1 Tf
0.2778 0 TD
(\))Tj
/F16 1 Tf
-33.5838 -1.2694 TD
[(is)-357.8(as)-357.8(in)-357.7(The)51.1(or)51.1(em)]TJ
/F2 1 Tf
7.7305 0 TD
(1.5.)Tj
-5.7214 -2.1826 TD
[(These)-532.4(b)-27.7(ounds)-532.4(are)]TJ
/F7 1 Tf
8.6391 0 TD
()Tj
/F2 1 Tf
1.6097 0 TD
0.5324 Tc
[(1i)532.4(f)]TJ
/F8 1 Tf
2.1482 0 TD
0 Tc
(y)Tj
/F2 1 Tf
1.1358 0 TD
[(=)-609.7(\(log)]TJ
/F8 1 Tf
3.2347 0 TD
()Tj
/F2 1 Tf
0.4166 0 TD
(\))Tj
/F14 1 Tf
7.9701 0 0 7.9701 332.4234 145.0301 Tm
(O)Tj
/F1 1 Tf
0.8356 0 TD
(\(1\))Tj
/F2 1 Tf
10.9091 0 0 10.9091 350.403 141.071 Tm
[(,)-582.2(and)]TJ
/F7 1 Tf
3.0036 0 TD
()Tj
/F8 1 Tf
1.6097 0 TD
(y)Tj
/F15 1 Tf
7.9701 0 0 7.9701 406.4687 145.0301 Tm
()Tj
/F14 1 Tf
0.8265 0 TD
()Tj
/F1 1 Tf
0.5869 0 TD
(\(1+)Tj
/F14 1 Tf
1.7709 0 TD
(o)Tj
/F1 1 Tf
0.5136 0 TD
(\(1\)\))Tj
/F2 1 Tf
10.9091 0 0 10.9091 456.3617 141.071 Tm
(if)Tj
/F8 1 Tf
1.1158 0 TD
(y)Tj
/F2 1 Tf
1.1358 0 TD
(=)Tj
-33.1948 -1.2694 TD
(exp\(\(log)Tj
/F8 1 Tf
3.7639 0 TD
()Tj
/F2 1 Tf
0.4166 0 TD
(\))Tj
/F14 1 Tf
7.9701 0 0 7.9701 168.6486 131.182 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 173.8242 127.2229 Tm
[(\))-471.2(with)-471.2(0)]TJ
/F7 1 Tf
4.2833 0 TD
()Tj
/F8 1 Tf
1.2853 0 TD
0.6207 Tc
(<)Tj
/F2 1 Tf
2.3432 0 TD
0 Tc
(1)Tj
/F8 1 Tf
0.5 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
[(2.)-858(The)-471.2(corresp)-27.7(onding)-471.1(result)-471.1(in)-471.2([4,)-505.6(Th.)-471.2(A1],)]TJ
ET
endstream
endobj
104 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F6 9 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F10 13 0 R
/F14 22 0 R
/F15 23 0 R
/F16 24 0 R
/F17 28 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
106 0 obj
<<
/Length 17600
>>
stream
BT
/F5 1 Tf
7.9701 0 0 7.9701 162.3741 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
[(AN)-377.7(UNCER)87.7(T)87.5(AINTY)-377.5(PRINCIPLE)-377.6(F)29.2(O)0(R)-377.8(ARITHMETIC)-377.4(SEQUENCES)]TJ
/F2 1 Tf
10.9091 0 0 10.9091 478.5006 726.9741 Tm
(27)Tj
-32.9726 -2.3745 TD
[(giv)27.8(es)-344.5(the)-344.6(w)27.8(eak)27.9(er)-344.6(b)-27.7(ound)]TJ
/F8 1 Tf
10.6088 0 TD
(y)Tj
/F15 1 Tf
7.9701 0 0 7.9701 240.272 705.0302 Tm
()Tj
/F1 1 Tf
0.8264 0 TD
(\(1+)Tj
/F14 1 Tf
1.7709 0 TD
(o)Tj
/F1 1 Tf
0.5135 0 TD
(\(1\)\))Tj
/F14 1 Tf
1.7709 0 TD
0.1155 Tc
(/)Tj
/F1 1 Tf
1.1182 0 TD
0 Tc
(\(1)Tj
/F15 1 Tf
0.9445 0 TD
()Tj
/F14 1 Tf
0.8264 0 TD
()Tj
/F1 1 Tf
0.587 0 TD
(\))Tj
/F2 1 Tf
10.9091 0 0 10.9091 314.4339 701.071 Tm
[(\(though)-344.5(our)-344.5(b)-27.7(ound)-344.5(is)-344.6(obtained)-344.4(there)]TJ
-17.9332 -1.2694 TD
[(assuming)-345.2(the)-345.3(Generalized)-345.2(Riemann)-345.3(Hyp)-27.7(othesis\).)-480.4(Our)-345.3(constrain)27.9(t)-345.4(o)0(n)-345.3(the)-345.4(small)]TJ
T*
[(primes)-456(dividing)]TJ
/F8 1 Tf
7.3373 0 TD
()Tj
/F2 1 Tf
0.8729 0 TD
[(is)-456.1(less)-456.2(restrictiv)27.9(e)-456.1(than)-456.1(the)-456.1(corresp)-27.7(onding)-456(condition)-456.1(there,)]TJ
-8.2101 -1.2694 TD
[(though)-462.3(our)-462.2(lo)-27.8(calization)-462.2(of)-462.3(the)]TJ
/F8 1 Tf
13.981 0 TD
(x)Tj
/F15 1 Tf
7.9701 0 0 7.9701 277.5542 657.8909 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 289.6827 659.5268 Tm
[(v)55.5(alues)-462.2(is)-462.3(w)27.8(orse)-462.3(\(in)-462.3([4])-462.2(the)]TJ
/F8 1 Tf
12.1822 0 TD
(x)Tj
/F15 1 Tf
7.9701 0 0 7.9701 428.814 657.8909 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 440.9424 659.5268 Tm
[(v)55.6(alues)-462.3(are)]TJ
-29.5298 -1.2694 TD
[(lo)-27.8(calized)-333.3(in)-333.3(in)27.9(terv)55.6(als)-333.3(\()]TJ
/F8 1 Tf
9.6472 0 TD
(x/)Tj
/F2 1 Tf
1.0715 0 TD
(2)Tj
/F8 1 Tf
0.5 0 TD
(,)Tj
/F2 1 Tf
0.4445 0 TD
(2)Tj
/F8 1 Tf
0.5 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
0.0001 Tc
(\)\).)Tj
-10.7256 -1.2694 TD
0 Tc
[(Theorem)-362.7(5.1)-362.8(omits)-362.7(a)-362.8(thin)-362.7(set)-362.8(of)-362.7(mo)-27.7(duli)]TJ
/F8 1 Tf
17.8051 0 TD
()Tj
/F2 1 Tf
0.7795 0 TD
[(ha)27.8(ving)-362.8(v)27.8(ery)-362.7(man)27.8(y)-362.8(small)-362.7(prime)]TJ
-20.5937 -1.2694 TD
[(factors.)-444.3(W)83.3(e)-333.3(next)-333.3(giv)27.8(e)-333.3(a)-333.3(w)27.7(eak)27.9(er)-333.4(v)55.6(arian)27.8(t)-333.3(whic)27.8(h)-333.4(includes)-333.2(all)-333.4(mo)-27.7(duli)]TJ
/F8 1 Tf
28.6584 0 TD
()Tj
/F2 1 Tf
0.4167 0 TD
(.)Tj
/F5 1 Tf
-27.0659 -2.1827 TD
[(Theorem)-377.6(5.2)]TJ
/F2 1 Tf
6.3022 0 TD
(.)Tj
/F16 1 Tf
1.1667 0 TD
-0.0511 Tc
[(Le)-51.1(t)]TJ
/F8 1 Tf
1.7261 0 TD
0 Tc
()Tj
/F16 1 Tf
0.7744 0 TD
[(b)51.1(e)-357.8(lar)51.2(ge)-357.8(and)-357.7(supp)51.2(ose)-357.8(that)]TJ
/F2 1 Tf
-3.9292 -2.6407 TD
(\(log)Tj
/F8 1 Tf
1.8473 0 TD
()Tj
/F2 1 Tf
0.4166 0 TD
(\))Tj
/F1 1 Tf
7.9701 0 0 7.9701 235.5497 569.868 Tm
(1+)Tj
/F14 1 Tf
1.3577 0 TD
()Tj
/F7 1 Tf
10.9091 0 0 10.9091 253.8092 565.3639 Tm
()Tj
/F8 1 Tf
1.0556 0 TD
(y)Tj
/F7 1 Tf
0.8039 0 TD
()Tj
/F2 1 Tf
1.0555 0 TD
(exp\()Tj
/F8 1 Tf
1.9167 0 TD
(\012)Tj
/F9 1 Tf
0.6184 0.9037 TD
()Tj
ET
324.21 575.686 20.454 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 324.1743 565.3639 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(/)Tj
/F9 1 Tf
0.9166 0.9037 TD
()Tj
ET
1 i
361.05 575.25 36.364 0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 360.9923 565.3639 Tm
[(log)-180.6(log)]TJ
/F8 1 Tf
2.9167 0 TD
()Tj
/F2 1 Tf
0.4166 0 TD
(\))Tj
/F16 1 Tf
-25.5343 -2.6408 TD
[(for)-412.6(a)-412.6(c)51.1(ertain)-412.5(absolute)-412.5(c)51.1(onstant)]TJ
/F8 1 Tf
13.9207 0 TD
0.4299 Tc
(\012>)Tj
/F2 1 Tf
2.1504 0 TD
0 Tc
(0,)Tj
/F16 1 Tf
1.2041 0 TD
[(and)-412.6(put)]TJ
/F8 1 Tf
3.7897 0 TD
(x)Tj
/F2 1 Tf
0.9486 0 TD
(=)Tj
/F8 1 Tf
1.155 0 TD
0.0358 Tc
(y)Tj
/F16 1 Tf
0.9427 0 TD
0 Tc
[(.)-624.6(Ther)51.2(e)-412.6(exist)-412.6(numb)51.1(ers)]TJ
/F8 1 Tf
-24.1112 -1.2694 TD
(x)Tj
/F15 1 Tf
7.9701 0 0 7.9701 125.0341 521.071 Tm
()Tj
/F16 1 Tf
10.9091 0 0 10.9091 135.9811 522.7068 Tm
[(in)-354(the)-354.1(interval)]TJ
/F2 1 Tf
6.5438 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
[(x,)-166.7(xy)]TJ
/F14 1 Tf
7.9701 0 0 7.9701 234.668 526.666 Tm
0.0296 Tc
(D/)Tj
/F1 1 Tf
1.6096 0 TD
0 Tc
[(log)-14.7(\(log)]TJ
/F14 1 Tf
3.3351 0 TD
0.0358 Tc
(y/)Tj
/F1 1 Tf
1.2694 0 TD
0 Tc
[(log)-191.8(log)]TJ
/F14 1 Tf
3.099 0 TD
()Tj
/F1 1 Tf
0.434 0 TD
(\))Tj
/F2 1 Tf
10.9091 0 0 10.9091 316.1455 522.7068 Tm
(\),)Tj
/F16 1 Tf
1.0215 0 TD
[(and)-354(inte)51.1(gers)]TJ
/F8 1 Tf
5.6531 0 TD
(a)Tj
/F15 1 Tf
7.9701 0 0 7.9701 394.7248 521.071 Tm
()Tj
/F16 1 Tf
10.9091 0 0 10.9091 405.6718 522.7068 Tm
[(c)51.1(oprime)-354(to)]TJ
/F8 1 Tf
4.9886 0 TD
()Tj
/F16 1 Tf
0.7708 0 TD
(such)Tj
-32.056 -1.2694 TD
(that)Tj
/F2 1 Tf
1.9621 -3.05 TD
(\()Tj
/F8 1 Tf
1.2222 0 TD
(x)Tj
/F1 1 Tf
7.9701 0 0 7.9701 159.7719 473.9497 Tm
(+)Tj
/F2 1 Tf
10.9091 0 0 10.9091 166.8573 475.5855 Tm
(;)Tj
/F8 1 Tf
0.4444 0 TD
[(,)-166.6(a)]TJ
/F1 1 Tf
7.9701 0 0 7.9701 186.8652 473.9497 Tm
(+)Tj
/F2 1 Tf
10.9091 0 0 10.9091 193.9507 475.5855 Tm
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F8 1 Tf
1.85 0.6765 TD
(y)Tj
/F15 1 Tf
7.9701 0 0 7.9701 227.1452 486.925 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
()Tj
/F12 1 Tf
4.9813 0 0 4.9813 237.4785 485.9287 Tm
(1)Tj
/F1 1 Tf
7.9701 0 0 7.9701 241.3669 486.925 Tm
(\()Tj
/F14 1 Tf
0.4132 0 TD
(,y)Tj
/F1 1 Tf
1.2901 0 TD
(\))Tj
ET
213.93 478.486 52.273 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 213.9337 468.1027 Tm
[(log)-180.6(log)-180.5(log)]TJ
/F8 1 Tf
4.375 0 TD
()Tj
/F16 1 Tf
2.5707 0.6859 TD
(and)Tj
/F2 1 Tf
3.6289 0 TD
(\()Tj
/F8 1 Tf
1.2223 0 TD
(x)Tj
/F15 1 Tf
7.9701 0 0 7.9701 348.8608 473.9497 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 355.9452 475.5855 Tm
(;)Tj
/F8 1 Tf
0.4445 0 TD
[(,)-166.6(a)]TJ
/F15 1 Tf
7.9701 0 0 7.9701 375.9542 473.9497 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 383.0386 475.5855 Tm
(\))Tj
/F7 1 Tf
0.6667 0 TD
0.2777 Tc
()Tj
/F8 1 Tf
2.6278 0.6765 TD
0 Tc
(y)Tj
/F15 1 Tf
7.9701 0 0 7.9701 424.7183 486.925 Tm
()Tj
/F14 1 Tf
0.8264 0 TD
()Tj
/F12 1 Tf
4.9813 0 0 4.9813 435.0516 485.9287 Tm
(1)Tj
/F1 1 Tf
7.9701 0 0 7.9701 438.94 486.925 Tm
(\()Tj
/F14 1 Tf
0.4132 0 TD
(,y)Tj
/F1 1 Tf
1.2903 0 TD
(\))Tj
ET
411.45 478.486 52.273 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 411.5069 468.1027 Tm
[(log)-180.6(log)-180.5(log)]TJ
/F8 1 Tf
4.375 0 TD
()Tj
0.5263 0.6859 TD
(,)Tj
/F16 1 Tf
-31.7328 -3.0931 TD
[(wher)51.1(e)]TJ
/F8 1 Tf
2.8512 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 154.7527 440.2062 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 159.4849 441.8431 Tm
(\()Tj
/F8 1 Tf
0.3889 0 TD
[(x,)-166.7(y)]TJ
/F2 1 Tf
1.5421 0 TD
[(\))-327.4(=)-327.3(\(log)-180.5(log)]TJ
/F8 1 Tf
5.1269 0 TD
(y)Tj
/F2 1 Tf
0.7682 0 TD
(+)Tj
/F8 1 Tf
1.0198 0 TD
(O)Tj
/F2 1 Tf
0.7906 0 TD
(\(1\)\))Tj
/F8 1 Tf
1.6667 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
[(\(log)-180.6(log)]TJ
/F8 1 Tf
3.3056 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\))Tj
/F16 1 Tf
0.3889 0 TD
[(.)-542.1(Her)51.2(e)]TJ
/F8 1 Tf
3.2678 0 TD
(D)Tj
/F16 1 Tf
1.2408 0 TD
[(is)-385.2(an)-385.1(absolute)-385(p)51.1(ositive)]TJ
-24.3072 -1.2694 TD
[(c)51.1(onstant)-357.7(which)-357.7(dep)51.1(ends)-357.7(only)-357.7(on)]TJ
/F8 1 Tf
14.1321 0 TD
()Tj
/F16 1 Tf
0.4663 0 TD
(.)Tj
/F2 1 Tf
-12.5893 -2.1826 TD
[(Theorem)-348.9(5.2)-349(should)-348.9(b)-27.8(e)-349(compared)-348.9(with)-348.9(Theorem)-348.9(A2)-349(of)-348.9([4].)-491.3(Our)-348.9(b)-27.8(ound)-348.9(is)]TJ
/F7 1 Tf
-2.0091 -1.2694 TD
()Tj
/F8 1 Tf
1.2858 0 TD
(y)Tj
/F15 1 Tf
7.9701 0 0 7.9701 138.5654 394.2954 Tm
()Tj
/F14 1 Tf
0.8265 0 TD
()Tj
/F1 1 Tf
0.587 0 TD
(\(1+)Tj
/F14 1 Tf
1.7708 0 TD
(o)Tj
/F1 1 Tf
0.5136 0 TD
(\(1\)\))Tj
/F2 1 Tf
10.9091 0 0 10.9091 186.3382 390.3363 Tm
(if)Tj
/F8 1 Tf
0.9215 0 TD
(y)Tj
/F2 1 Tf
0.8119 0 TD
[(=)-285.8(exp\(\(log)]TJ
/F8 1 Tf
4.8274 0 TD
()Tj
/F2 1 Tf
0.4167 0 TD
(\))Tj
/F14 1 Tf
7.9701 0 0 7.9701 266.6979 394.2954 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 271.8745 390.3363 Tm
[(\))-338.1(with)-338.1(0)]TJ
/F8 1 Tf
3.7953 0 TD
0.2857 Tc
[(<)-113.3(<)]TJ
/F2 1 Tf
2.9631 0 TD
0 Tc
(1)Tj
/F8 1 Tf
0.5 0 TD
(/)Tj
/F2 1 Tf
0.5 0 TD
[(2.)-458.8(The)-338(corresp)-27.7(onding)-338(result)]TJ
-21.7903 -1.2694 TD
[(in)-331.6([4,)-331.9(Th.)-331.5(A2],)-331.8(giv)27.8(es)-331.5(a)-331.6(w)27.8(eak)27.8(er)-331.5(b)-27.7(ound)]TJ
/F8 1 Tf
16.2449 0 TD
(y)Tj
/F15 1 Tf
7.9701 0 0 7.9701 301.7564 380.4473 Tm
()Tj
/F1 1 Tf
0.8265 0 TD
(\(3+)Tj
/F14 1 Tf
1.7709 0 TD
(o)Tj
/F1 1 Tf
0.5135 0 TD
(\(1\)\))Tj
/F14 1 Tf
1.7709 0 TD
0.1155 Tc
(/)Tj
/F1 1 Tf
1.1183 0 TD
0 Tc
(\(1+)Tj
/F14 1 Tf
1.7709 0 TD
()Tj
/F1 1 Tf
0.587 0 TD
(\))Tj
/F2 1 Tf
10.9091 0 0 10.9091 372.1604 376.4882 Tm
[(,)-331.9(though)-331.5(our)-331.6(lo)-27.8(calization)]TJ
-23.2248 -1.2694 TD
[(of)-333.3(the)]TJ
/F8 1 Tf
2.8611 0 TD
(x)Tj
/F15 1 Tf
7.9701 0 0 7.9701 156.2461 361.0033 Tm
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 166.9679 362.6401 Tm
[(v)55.5(alues)-333.2(is)-333.3(again)-333.3(m)27.8(u)0(c)27.8(h)-333.3(w)27.8(orse.)]TJ
-2.4063 -1.2694 TD
[(T)83.3(o)-321(pro)27.8(v)27.8(e)-321.1(Theorems)-320.9(1.5,)-323.5(5.1)-321(and)-321(5.2)-321.1(w)27.8(e)-321(require)-321(kno)27.8(wledge)-321(of)-321.1(the)-320.9(distribu-)]TJ
-2.0091 -1.2694 TD
[(tion)-307(of)-307(primes)-307(in)-306.9(certain)-307(arithmetic)-306.9(progressions.)-435.6(W)83.4(e)-307.1(b)-27.7(egin)-307(b)27.8(y)-307(describing)-306.9(suc)27.8(h)]TJ
T*
[(a)-294.8(result,)-302.4(whic)27.8(h)-294.9(will)-294.7(b)-27.8(e)-294.8(deduced)-294.7(as)-294.9(a)-294.8(consequence)-294.7(of)-294.8(a)-294.8(theorem)-294.7(of)-294.8(Gallagher)-294.7([5].)]TJ
2.0091 -1.2694 TD
-0.0834 Tc
[(Fo)-83.4(r)-558.8(1)]TJ
/F7 1 Tf
2.9508 0 TD
0 Tc
()Tj
/F8 1 Tf
1.2921 0 TD
(j)Tj
/F7 1 Tf
0.9835 0 TD
()Tj
/F8 1 Tf
1.2921 0 TD
(J)Tj
/F2 1 Tf
1.165 0 TD
[(:=)-514.4([log)]TJ
/F8 1 Tf
3.306 0 TD
0.044 Tc
(z/)Tj
/F2 1 Tf
1.009 0 TD
0 Tc
[(\(2)-166.6(log)-180.6(2\)],)-510.8(consider)-475.2(the)-475.3(dy)27.8(adic)-475.2(in)27.8(terv)55.6(als)]TJ
/F8 1 Tf
17.8314 0 TD
(I)Tj
/F14 1 Tf
7.9701 0 0 7.9701 470.931 305.611 Tm
(j)Tj
/F2 1 Tf
10.9091 0 0 10.9091 480.9246 307.2478 Tm
(=)Tj
-33.1948 -1.2694 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
0.044 Tc
(z/)Tj
/F2 1 Tf
1.009 0 TD
0 Tc
(2)Tj
/F14 1 Tf
7.9701 0 0 7.9701 139.5039 297.3589 Tm
(j)Tj
/F8 1 Tf
10.9091 0 0 10.9091 143.8864 293.3997 Tm
0.1667 Tc
[(,z)122.8(/)]TJ
/F2 1 Tf
1.4534 0 TD
0 Tc
(2)Tj
/F14 1 Tf
7.9701 0 0 7.9701 165.1965 297.3589 Tm
(j)Tj
/F15 1 Tf
0.4874 0 TD
()Tj
/F1 1 Tf
0.8264 0 TD
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 180.3995 293.3997 Tm
[(].)-581.3(Let)]TJ
/F8 1 Tf
2.9742 0 TD
(P)Tj
/F14 1 Tf
7.9701 0 0 7.9701 219.8486 291.7629 Tm
(j)Tj
/F2 1 Tf
10.9091 0 0 10.9091 228.3656 293.3997 Tm
[(denote)-378.9(a)-378.9(subset)-378.9(of)-378.9(the)-378.9(primes)-378.9(in)]TJ
/F8 1 Tf
14.6999 0 TD
(I)Tj
/F14 1 Tf
7.9701 0 0 7.9701 393.5233 291.7629 Tm
(j)Tj
/F2 1 Tf
10.9091 0 0 10.9091 397.9059 293.3997 Tm
[(,)-390.3(and)-378.9(let)]TJ
/F8 1 Tf
4.1483 0 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 449.3779 291.7629 Tm
(j)Tj
/F2 1 Tf
10.9091 0 0 10.9091 457.8949 293.3997 Tm
(denote)Tj
-31.0837 -1.2694 TD
[(the)-331.5(cardinalit)27.9(y)-331.6(o)0(f)]TJ
/F8 1 Tf
7.8585 0 TD
(P)Tj
/F14 1 Tf
7.9701 0 0 7.9701 211.5327 277.9148 Tm
(j)Tj
/F2 1 Tf
10.9091 0 0 10.9091 215.9153 279.5516 Tm
[(.)-443.8(W)83.3(e)-331.5(let)]TJ
/F7 1 Tf
3.8847 0 TD
(Q)Tj
/F2 1 Tf
1.1482 0 TD
[(denote)-331.5(the)-331.5(set)-331.5(of)-331.6(in)27.9(tegers)]TJ
/F8 1 Tf
11.3383 0 TD
(q)Tj
/F2 1 Tf
0.8139 0 TD
[(with)-331.5(the)-331.5(follo)27.8(wing)]TJ
-26.0873 -1.2694 TD
[(prop)-27.8(ert)27.9(y:)]TJ
/F8 1 Tf
4.6616 0 TD
(q)Tj
/F2 1 Tf
0.9132 0 TD
(=)Tj
/F9 1 Tf
1.2087 0.7499 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 203.1053 270.7307 Tm
(J)Tj
0 -1.0355 TD
(j)Tj
/F1 1 Tf
0.4874 0 TD
(=1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 220.1265 265.7036 Tm
(q)Tj
/F14 1 Tf
7.9701 0 0 7.9701 224.9963 264.0667 Tm
(j)Tj
/F2 1 Tf
10.9091 0 0 10.9091 234.0174 265.7036 Tm
[(and)-425.2(eac)27.8(h)]TJ
/F8 1 Tf
4.3783 0 TD
(q)Tj
/F14 1 Tf
7.9701 0 0 7.9701 286.6501 264.0667 Tm
(j)Tj
/F2 1 Tf
10.9091 0 0 10.9091 295.6712 265.7036 Tm
[(is)-425.3(the)-425.2(pro)-27.7(duct)-425.2(of)-425.2(exactly)-425.2([)]TJ
/F8 1 Tf
11.8013 0 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 430.6312 264.0667 Tm
(j)Tj
/F8 1 Tf
10.9091 0 0 10.9091 435.0127 265.7036 Tm
(/)Tj
/F2 1 Tf
0.5 0 TD
[(2])-425.3(distinct)]TJ
-29.4862 -1.2694 TD
[(primes)-300.1(in)]TJ
/F8 1 Tf
4.3311 0 TD
(P)Tj
/F14 1 Tf
7.9701 0 0 7.9701 173.0511 250.2187 Tm
(j)Tj
/F2 1 Tf
10.9091 0 0 10.9091 177.4336 251.8556 Tm
[(.)-433.4(I)0(t)-300.3(i)0(s)-300.2(clear)-300.2(that)-300.2(all)-300.2(the)-300.2(elemen)27.8(ts)-300.2(of)]TJ
/F7 1 Tf
15.4327 0 TD
(Q)Tj
/F2 1 Tf
1.1169 0 TD
[(are)-300.2(squarefree)-300.2(and)-300.2(lie)-300.3(b)-27.7(elo)27.7(w)]TJ
/F8 1 Tf
-21.9244 -1.2694 TD
(Q)Tj
/F2 1 Tf
1.0683 0 TD
(:=)Tj
/F8 1 Tf
1.3333 0 TD
(z)Tj
/F9 1 Tf
8.4384 0 0 7.6712 150.5524 248.3198 Tm
()Tj
/F10 1 Tf
4.9813 0 0 4.9813 159.7609 240.1504 Tm
(j)Tj
/F14 1 Tf
7.9701 0 0 7.9701 164.8946 242.4916 Tm
( )Tj
/F10 1 Tf
4.9813 0 0 4.9813 169.7574 241.4954 Tm
(j)Tj
/F14 1 Tf
7.9701 0 0 7.9701 173.4794 242.4916 Tm
(/)Tj
/F1 1 Tf
0.5313 0 TD
(2)Tj
/F2 1 Tf
10.9091 0 0 10.9091 182.4458 238.0074 Tm
[(,)-333.3(and)-333.3(that)]TJ
/F7 1 Tf
4.7222 0 TD
(|Q|)Tj
/F2 1 Tf
1.65 0 TD
(=)Tj
/F9 1 Tf
1.0555 0.75 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 273.7793 243.0346 Tm
(J)Tj
0 -1.0355 TD
(j)Tj
/F1 1 Tf
0.4873 0 TD
(=1)Tj
/F9 1 Tf
10.9091 0 0 10.9091 290.7995 246.8443 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 302.386 243.1721 Tm
( )Tj
/F10 1 Tf
4.9813 0 0 4.9813 307.2488 242.1758 Tm
(j)Tj
/F1 1 Tf
7.9701 0 0 7.9701 295.7998 233.9208 Tm
([)Tj
/F14 1 Tf
0.2951 0 TD
( )Tj
/F10 1 Tf
4.9813 0 0 4.9813 303.0147 232.9245 Tm
(j)Tj
/F14 1 Tf
7.9701 0 0 7.9701 306.7368 233.9208 Tm
(/)Tj
/F1 1 Tf
0.5313 0 TD
(2])Tj
/F9 1 Tf
10.9091 0 0 10.9091 317.5582 246.8443 Tm
()Tj
/F2 1 Tf
0.4583 -0.81 TD
(.)Tj
-16.6687 -1.2694 TD
[(There)-409.6(is)-409.7(a)-409.7(constan)27.8(t)]TJ
/F8 1 Tf
9.0693 0 TD
(c)Tj
/F1 1 Tf
7.9701 0 0 7.9701 244.3766 222.5226 Tm
(1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 253.5781 224.1593 Tm
[(suc)27.8(h)-409.7(that)-409.6(at)-409.6(most)-409.7(one)-409.6(primitiv)27.8(e)]TJ
/F8 1 Tf
14.6666 0 TD
(L)Tj
/F2 1 Tf
0.6805 0 TD
[(-function)-409.5(with)]TJ
-27.7018 -1.2694 TD
[(mo)-27.7(dulus)]TJ
/F8 1 Tf
4.0552 0 TD
(q)Tj
/F2 1 Tf
0.8374 0 TD
[(b)-27.8(e)0(t)27.8(w)27.8(een)]TJ
/F7 1 Tf
3.8829 0.8467 TD
()Tj
ET
223.65 220.006 7.89 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 223.6224 210.3113 Tm
(T)Tj
/F2 1 Tf
1.0784 0 TD
(and)Tj
/F8 1 Tf
1.9662 0 TD
(T)Tj
/F2 1 Tf
1.0784 0 TD
[(has)-355.1(a)-355.2(zero)-355.1(in)-355.1(the)-355.1(region)]TJ
/F8 1 Tf
10.753 0 TD
0.35 Tc
(>)Tj
/F2 1 Tf
2.0133 0 TD
0 Tc
(1)Tj
/F7 1 Tf
0.7367 0 TD
()Tj
/F8 1 Tf
1.0145 0 TD
(c)Tj
/F1 1 Tf
7.9701 0 0 7.9701 431.6952 208.6745 Tm
(1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 436.4274 210.3113 Tm
(/)Tj
/F2 1 Tf
0.6667 0 TD
(log)Tj
/F8 1 Tf
1.4583 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
[(,)-360.6(and)]TJ
/F7 1 Tf
-31.7231 -1.2694 TD
(|)Tj
/F8 1 Tf
0.2778 0 TD
(t)Tj
/F7 1 Tf
0.3611 0 TD
0.2777 Tc
(|)Tj
/F8 1 Tf
1.6111 0 TD
0 Tc
(T)Tj
/F2 1 Tf
0.7233 0 TD
[(.)-433.1(F)83.4(urther)-299.1(if)-299.2(this)-299.1(exceptional)-299.1(Siegel)-299(zero)-299.1(exists)-299.1(then)-299.2(it)-299.2(is)-299.2(real,)-305.9(simple)-299.1(and)]TJ
-2.9732 -1.2694 TD
[(unique)-409.7(\(see)-409.8(Chapter)-409.7(14)-409.8(in)-409.9([2]\).)-673.9(W)83.3(e)-409.8(call)-409.8(the)-409.8(mo)-27.7(dulus)-409.8(of)-409.8(suc)27.8(h)-409.8(a)0(n)-409.8(exceptional)]TJ
T*
[(c)27.7(haracter)-461.2(a)-461.4(Siegel)-461.3(mo)-27.7(dulus.)-828.5(Belo)27.8(w)]TJ
/F8 1 Tf
16.3101 0 TD
(Q)Tj
/F2 1 Tf
1.252 0 TD
[(there)-461.3(are)]TJ
/F7 1 Tf
4.4838 0 TD
()Tj
/F2 1 Tf
1.4911 0 TD
[(log)-180.6(log)]TJ
/F8 1 Tf
2.9167 0 TD
(Q)Tj
/F2 1 Tf
1.2519 0 TD
[(Siegel)-461.3(mo)-27.7(duli.)]TJ
-27.7057 -1.2694 TD
[(Denote)-403.3(these)-403.4(b)27.8(y)]TJ
/F8 1 Tf
7.5909 0 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 206.9978 153.2822 Tm
(1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 211.731 154.919 Tm
0.1666 Tc
(,)Tj
/F1 1 Tf
7.9701 0 0 7.9701 221.9676 153.2822 Tm
0 Tc
(2)Tj
/F8 1 Tf
10.9091 0 0 10.9091 226.6999 154.919 Tm
0.1667 Tc
[(,...)-166.7(,)0.1()]TJ
/F14 1 Tf
7.9701 0 0 7.9701 258.15 153.1417 Tm
0 Tc
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 262.1071 154.919 Tm
[(,)-420.9(and)-403.5(for)-403.4(eac)27.8(h)-403.4(select)-403.4(a)-403.5(prime)-403.4(divisor)]TJ
/F8 1 Tf
16.57 0 TD
(v)Tj
/F1 1 Tf
7.9701 0 0 7.9701 448.1594 153.2822 Tm
(1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 452.8917 154.919 Tm
0.1667 Tc
(,...,v)Tj
/F14 1 Tf
7.9701 0 0 7.9701 482.4219 153.1417 Tm
0 Tc
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 486.3791 154.919 Tm
(.)Tj
-33.6948 -1.2694 TD
[(Assume)-371.8(none)-371.8(of)]TJ
/F8 1 Tf
7.3489 0 TD
(v)Tj
/F1 1 Tf
7.9701 0 0 7.9701 204.257 139.4351 Tm
(1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 208.9893 141.071 Tm
0.1667 Tc
(,...v)Tj
/F14 1 Tf
7.9701 0 0 7.9701 233.6707 139.2935 Tm
0 Tc
()Tj
/F2 1 Tf
10.9091 0 0 10.9091 241.6847 141.071 Tm
[(b)-27.7(elongs)-371.8(to)]TJ
/F7 1 Tf
4.8881 0 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 302.2824 145.0301 Tm
(J)Tj
0 -0.8801 TD
(j)Tj
/F1 1 Tf
0.4873 0 TD
(=1)Tj
/F8 1 Tf
10.9091 0 0 10.9091 317.4854 141.071 Tm
(P)Tj
/F14 1 Tf
7.9701 0 0 7.9701 324.4892 139.4351 Tm
(j)Tj
/F2 1 Tf
10.9091 0 0 10.9091 328.8708 141.071 Tm
[(,)-381.5(whic)27.8(h)-371.8(guaran)27.9(tees)-371.8(that)-371.7(there)-371.8(are)]TJ
-19.2565 -1.2694 TD
[(no)-333.3(Siegel)-333.3(zeros)-333.3(for)-333.3(an)27.8(y)-333.4(o)0(f)-333.2(the)-333.4(mo)-27.7(duli)]TJ
/F8 1 Tf
16.3722 0 TD
(d)Tj
/F2 1 Tf
0.8538 0 TD
(where)Tj
/F8 1 Tf
2.8917 0 TD
(d)Tj
/F2 1 Tf
0.8538 0 TD
[(is)-333.3(a)-333.3(divisor)-333.3(of)-333.3(some)]TJ
/F8 1 Tf
8.7417 0 TD
(q)Tj
/F7 1 Tf
0.76 0 TD
0.2778 Tc
(Q)Tj
/F2 1 Tf
1.7612 0 TD
0 Tc
(.)Tj
ET
endstream
endobj
107 0 obj
<<
/ProcSet [/PDF /Text ]
/Font <<
/F1 4 0 R
/F2 5 0 R
/F5 8 0 R
/F7 10 0 R
/F8 11 0 R
/F9 12 0 R
/F10 13 0 R
/F12 15 0 R
/F14 22 0 R
/F15 23 0 R
/F16 24 0 R
>>
/ExtGState <<
/GS1 17 0 R
>>
>>
endobj
109 0 obj
<<
/Length 18813
>>
stream
BT
/F2 1 Tf
10.9091 0 0 10.9091 118.7995 726.9739 Tm
0 g
/GS1 gs
0 Tc
0 Tw
(28)Tj
/F5 1 Tf
7.9701 0 0 7.9701 202.119 726.9739 Tm
[(ANDREW)-377.5(GRANVILLE)-377.4(AND)-377.7(K)0(.)-377.7(SOUND)29.4(ARARAJAN)]TJ
10.9091 0 0 10.9091 140.7173 701.071 Tm
[(Pr)22.3(oposition)-484.7(5.3)]TJ
/F2 1 Tf
7.9626 0 TD
(.)Tj
/F16 1 Tf
1.7783 0 TD
[(Supp)51.2(ose)-451.6(that)]TJ
/F2 1 Tf
6.0397 0 TD
(exp\()Tj
/F7 1 Tf
1.9167 0.755 TD
()Tj
ET
342.81 709.726 22.144 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 342.8693 701.071 Tm
(log)Tj
/F8 1 Tf
1.4583 0 TD
(x)Tj
/F2 1 Tf
0.5716 0 TD
(\))Tj
/F7 1 Tf
0.8365 0 TD
()Tj
/F8 1 Tf
1.2255 0 TD
(Q)Tj
/F7 1 Tf
1.2382 0 TD
()Tj
/F8 1 Tf
1.2255 0 TD
(x)Tj
/F14 1 Tf
7.9701 0 0 7.9701 420.6187 705.0302 Tm
(b)Tj
/F16 1 Tf
10.9091 0 0 10.9091 429.6657 701.071 Tm
[(wher)51.2(e)]TJ
/F8 1 Tf
2.9176 0 TD
(b)Tj
/F16 1 Tf
0.8807 0 TD
[(is)-451.5(a)]TJ
-32.2944 -1.2694 TD
[(p)51.1(ositive)-357.7(absolute)-357.7(c)51.1(onstant)]TJ
/F2 1 Tf
11.0655 0 TD
(,)Tj
/F16 1 Tf
0.6355 0 TD
[(and)-357.8(let)]TJ
/F8 1 Tf
3.3478 0 TD
(x)Tj
/F2 1 Tf
0.7382 0 TD
(exp\()Tj
/F7 1 Tf
1.9167 0 TD
()Tj
0.7778 0.755 TD
()Tj
ET
1 i
329.49 695.49 22.144 0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 329.5063 687.2229 Tm
(log)Tj
/F8 1 Tf
1.4583 0 TD
(x)Tj
/F2 1 Tf
0.5716 0 TD
(\))Tj
/F7 1 Tf
0.6667 0 TD
()Tj
/F8 1 Tf
1.0555 0 TD
(h)Tj
/F7 1 Tf
0.8539 0 TD
()Tj
/F8 1 Tf
1.0556 0 TD
(x)Tj
/F16 1 Tf
0.5715 0 TD
[(.)-460(Then)]TJ
/F2 1 Tf
-23.857 -2.0283 TD
(1)Tj
ET
132.45 660.646 14.97 -0.436 re
f
BT
/F7 1 Tf
10.9091 0 0 10.9091 132.4872 650.2327 Tm
(|Q|)Tj
/F9 1 Tf
1.6884 1.6359 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 150.4707 644.4514 Tm
(q)Tj
/F15 1 Tf
0.5103 0 TD
(Q)Tj
/F2 1 Tf
10.9091 0 0 10.9091 172.9275 657.7156 Tm
(max)Tj
/F1 1 Tf
7.9701 0 0 7.9701 168.9176 649.9198 Tm
(\()Tj
/F14 1 Tf
0.4131 0 TD
(a,q)Tj
/F1 1 Tf
1.3697 0 TD
(\)=1)Tj
/F9 1 Tf
10.9091 0 0 10.9091 199.0595 673.5333 Tm
()Tj
0 -0.6 TD
()Tj
T*
()Tj
T*
()Tj
/F8 1 Tf
0.4429 1.0266 TD
()Tj
/F2 1 Tf
0.5914 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(x)Tj
/F2 1 Tf
0.7937 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
(h)Tj
/F2 1 Tf
0.5762 0 TD
(;)Tj
/F8 1 Tf
0.4445 0 TD
0.0359 Tc
[(q,)-130.8(a)]TJ
/F2 1 Tf
1.4553 0 TD
0 Tc
(\))Tj
/F7 1 Tf
0.611 0 TD
()Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
0.5915 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(;)Tj
/F8 1 Tf
0.4445 0 TD
0.0359 Tc
[(q,)-130.7(a)]TJ
/F2 1 Tf
1.4553 0 TD
0 Tc
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1 0 TD
(h/\015)Tj
/F2 1 Tf
1.7303 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
ET
203.85 660.646 162.701 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 268.9305 650.2327 Tm
(h/\015)Tj
/F2 1 Tf
1.7303 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
/F9 1 Tf
6.4604 2.1359 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
T*
()Tj
/F7 1 Tf
0.6111 0.35 TD
()Tj
/F2 1 Tf
1.2777 0 TD
(exp)Tj
/F9 1 Tf
1.6944 1.11 TD
( )Tj
/F7 1 Tf
0.8195 -1.11 TD
()Tj
/F8 1 Tf
1 0 TD
()Tj
/F7 1 Tf
0.753 1.4315 TD
()Tj
ET
444.09 673.726 22.144 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 444.0329 665.0959 Tm
(log)Tj
/F8 1 Tf
1.4583 0 TD
(x)Tj
ET
434.97 660.646 31.235 -0.436 re
f
BT
/F7 1 Tf
10.9091 0 0 10.9091 435.2827 658.0424 Tm
()Tj
ET
444.33 658.486 21.463 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 444.3736 649.8063 Tm
(log)Tj
/F8 1 Tf
1.4583 0 TD
(z)Tj
/F9 1 Tf
0.6499 1.835 TD
(\012)Tj
/F8 1 Tf
0.5973 -1.11 TD
(,)Tj
/F16 1 Tf
-32.5497 -3.0958 TD
[(wher)51.1(e)]TJ
/F8 1 Tf
2.8238 0 TD
()Tj
/F16 1 Tf
1.0012 0 TD
[(is)-357.8(a)-357.7(p)51.1(ositive)-357.8(absolute)-357.6(c)51(onstant.)]TJ
-1.8159 -2.1827 TD
[(Pr)51.2(o)51.1(o)0(f)]TJ
/F2 1 Tf
2.3267 0 TD
0.8889 Tc
[(.I)888.9(f)555.6(\()]TJ
/F8 1 Tf
2.5555 0 TD
0 Tc
[(a,)-166.7(q)]TJ
/F2 1 Tf
1.4553 0 TD
[(\))-277.8(=)-277.7(1)-333.3(then)-333.3(using)-333.2(the)-333.4(orthogonalit)28(y)-333.3(o)0(f)-333.3(c)27.7(haracters,)-333.2(w)27.8(e)-333.4(ha)27.8(v)27.8(e)]TJ
/F8 1 Tf
-8.2385 -3.0507 TD
()Tj
/F2 1 Tf
0.5914 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(x)Tj
/F2 1 Tf
0.7937 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
(h)Tj
/F2 1 Tf
0.5762 0 TD
(;)Tj
/F8 1 Tf
0.4445 0 TD
0.0359 Tc
[(q,)-130.8(a)]TJ
/F2 1 Tf
1.4553 0 TD
0 Tc
(\))Tj
/F7 1 Tf
0.6111 0 TD
()Tj
/F8 1 Tf
1 0 TD
()Tj
/F2 1 Tf
0.5914 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(;)Tj
/F8 1 Tf
0.4445 0 TD
0.0359 Tc
[(q,)-130.7(a)]TJ
/F2 1 Tf
1.4553 0 TD
0.1667 Tc
(\)=)Tj
2.3168 0.6765 TD
0 Tc
(1)Tj
ET
250.05 569.806 20.883 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 250.0424 559.3694 Tm
(\015)Tj
/F2 1 Tf
0.6542 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
/F9 1 Tf
1.9336 1.636 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 273.9387 553.056 Tm
(\012)Tj
/F1 1 Tf
1.7223 0 TD
[(\(mo)-29.4(d)]TJ
/F14 1 Tf
2.8039 0 TD
(q)Tj
/F1 1 Tf
0.5103 0 TD
(\))Tj
ET
319.17 576.766 21.077 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 319.191 566.8523 Tm
()Tj
/F2 1 Tf
0.6257 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(a)Tj
/F2 1 Tf
0.5286 0 TD
(\))Tj
/F14 1 Tf
7.9701 0 0 7.9701 342.0862 580.4892 Tm
(x)Tj
/F1 1 Tf
0.598 0 TD
(+)Tj
/F14 1 Tf
0.8265 0 TD
(h)Tj
/F9 1 Tf
10.9091 0 0 10.9091 342.3233 577.2164 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 347.8187 554.3063 Tm
(x)Tj
/F8 1 Tf
10.9091 0 0 10.9091 360.1375 566.8523 Tm
()Tj
/F2 1 Tf
0.6257 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5032 0 TD
[(\))-166.6(log)]TJ
/F8 1 Tf
2.0139 0 TD
(p)Tj
/F2 1 Tf
-14.6777 -3.77 TD
(=)Tj
1.7612 0.6765 TD
(1)Tj
ET
250.05 528.646 20.883 -0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 250.0424 518.2426 Tm
(\015)Tj
/F2 1 Tf
0.6542 0 TD
(\()Tj
/F8 1 Tf
0.389 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
/F9 1 Tf
1.5012 1.636 TD
()Tj
/F10 1 Tf
4.9813 0 0 4.9813 275.1342 514.3243 Tm
(x)Tj
/F11 1 Tf
0.7916 0 TD
()Tj
/F10 1 Tf
1.0834 0 TD
(p)Tj
/F11 1 Tf
0.7398 0 TD
()Tj
/F10 1 Tf
1.0834 0 TD
(x)Tj
/F12 1 Tf
0.7916 0 TD
(+)Tj
/F10 1 Tf
1.0278 0 TD
(h)Tj
/F12 1 Tf
-4.6872 -1.3594 TD
(\()Tj
/F10 1 Tf
0.5416 0 TD
-0.0001 Tc
(p,q)Tj
/F12 1 Tf
1.8928 0 TD
-0.0002 Tc
(\)=1)Tj
/F2 1 Tf
10.9091 0 0 10.9091 309.7554 525.7255 Tm
0 Tc
(log)Tj
/F8 1 Tf
1.4583 0 TD
(p)Tj
/F2 1 Tf
0.7254 0 TD
(+)Tj
/F8 1 Tf
1 0 TD
(O)Tj
/F9 1 Tf
0.7905 1.11 TD
( )Tj
/F2 1 Tf
1.414 -0.4335 TD
(1)Tj
ET
360.81 528.21 20.883 0.436 re
f
BT
/F8 1 Tf
10.9091 0 0 10.9091 360.8209 518.2426 Tm
(\015)Tj
/F2 1 Tf
0.6542 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.4824 0 TD
(\))Tj
/F9 1 Tf
0.9049 1.636 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 384.7173 512.3726 Tm
(\012)Tj
/F15 1 Tf
0.6597 0 TD
()Tj
/F1 1 Tf
T*
(=)Tj
/F14 1 Tf
0.8264 0 TD
(\012)Tj
/F12 1 Tf
4.9813 0 0 4.9813 401.8202 511.3763 Tm
(0)Tj
/F9 1 Tf
10.9091 0 0 10.9091 407.5258 538.2715 Tm
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 412.9803 539.3624 Tm
(x)Tj
/F1 1 Tf
0.5981 0 TD
(+)Tj
/F14 1 Tf
0.8264 0 TD
(h)Tj
/F9 1 Tf
10.9091 0 0 10.9091 413.2184 536.0897 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 418.7138 513.1806 Tm
(x)Tj
/F8 1 Tf
10.9091 0 0 10.9091 431.0327 525.7255 Tm
()Tj
/F2 1 Tf
0.6257 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
[(\))-166.7(log)]TJ
/F8 1 Tf
2.014 0 TD
(p)Tj
/F9 1 Tf
0.5031 1.15 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
0.3333 1.16 TD
(\012)Tj
/F8 1 Tf
0.5972 -1.11 TD
(.)Tj
/F2 1 Tf
-33.5866 -3.5177 TD
[(By)-333.4(the)-333.3(prime)-333.2(n)27.8(u)0(m)27.7(b)-27.7(er)-333.4(theorem)-333.2(the)-333.4(rst)-333.2(term)-333.3(ab)-27.8(o)27.9(v)27.8(e)-333.4(i)0(s)]TJ
/F8 1 Tf
9.934 -2.1215 TD
(h/\015)Tj
/F2 1 Tf
1.7303 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
/F7 1 Tf
0.3889 0 TD
({)Tj
/F2 1 Tf
0.5 0 TD
0.2222 Tc
(1+)Tj
/F8 1 Tf
1.7222 0 TD
0 Tc
(O)Tj
/F2 1 Tf
0.7906 0 TD
(\(exp\()Tj
/F7 1 Tf
2.3055 0 TD
()Tj
/F8 1 Tf
0.7778 0 TD
(c)Tj
/F9 1 Tf
0.4328 0.9038 TD
()Tj
ET
341.97 474.526 22.144 -0.436 re
f
BT
/F2 1 Tf
10.9091 0 0 10.9091 341.9258 464.2072 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(x)Tj
/F2 1 Tf
0.5715 0 TD
(\)\))Tj
/F7 1 Tf
0.7778 0 TD
(})Tj
/F8 1 Tf
0.5 0 TD
(,)Tj
/F2 1 Tf
-23.7609 -2.1215 TD
[(whic)27.8(h)-372.7(has)-372.6(an)-372.6(acceptable)-372.5(error)-372.6(term.)-562.3(W)83.3(e)-372.6(no)27.8(w)-372.7(f)0(o)-27.8(cus)-372.6(on)-372.6(estimating)-372.6(the)-372.6(second)]TJ
0 -1.2694 TD
[(term)-333.3(on)-333.3(a)27.8(v)27.8(erage.)]TJ
2.0091 -1.2694 TD
[(Belo)27.9(w)-313(the)-313(sup)-27.7(erscript)]TJ
/F7 1 Tf
9.7752 0 TD
()Tj
/F2 1 Tf
0.813 0 TD
[(will)-313(indicate)-312.9(a)-313.1(restriction)-312.9(to)-313(primitiv)27.9(e)-313(c)27.7(haracters.)]TJ
-12.5973 -1.2694 TD
[(Observ)27.8(e)-333.3(that)]TJ
/F9 1 Tf
1.6391 -1.8507 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 134.0632 355.6134 Tm
(\012)Tj
/F15 1 Tf
0.6598 0 TD
()Tj
/F1 1 Tf
T*
(=)Tj
/F14 1 Tf
0.8264 0 TD
(\012)Tj
/F12 1 Tf
4.9813 0 0 4.9813 151.1661 354.6172 Tm
(0)Tj
/F9 1 Tf
10.9091 0 0 10.9091 156.8727 381.5124 Tm
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 162.3273 382.6032 Tm
(x)Tj
/F1 1 Tf
0.5981 0 TD
(+)Tj
/F14 1 Tf
0.8264 0 TD
(h)Tj
/F9 1 Tf
10.9091 0 0 10.9091 162.5644 379.3305 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 168.0598 356.4214 Tm
(x)Tj
/F8 1 Tf
10.9091 0 0 10.9091 180.3786 368.9673 Tm
()Tj
/F2 1 Tf
0.6257 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
[(\))-166.6(log)]TJ
/F8 1 Tf
2.0139 0 TD
(p)Tj
/F9 1 Tf
0.5031 1.15 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F2 1 Tf
0.5 0.05 TD
(=)Tj
/F9 1 Tf
0.9445 0.95 TD
()Tj
/F10 1 Tf
4.9813 0 0 4.9813 243.3166 357.4127 Tm
(d)Tj
/F11 1 Tf
0.7396 0 TD
(|)Tj
/F10 1 Tf
0.4584 0 TD
(q)Tj
-1.5034 -1.3594 TD
-0.0001 Tc
(d>)Tj
/F12 1 Tf
1.823 0 TD
0 Tc
(1)Tj
/F9 1 Tf
10.9091 0 0 10.9091 280.2541 379.3305 Tm
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 296.011 377.2662 Tm
( )Tj
/F14 1 Tf
-3.4346 -2.7722 TD
(\012)Tj
/F1 1 Tf
1.7222 0 TD
0.0001 Tc
[(\(mo)-29.3(d)]TJ
/F14 1 Tf
2.8039 0 TD
0 Tc
(d)Tj
/F1 1 Tf
0.5466 0 TD
(\))Tj
/F9 1 Tf
10.9091 0 0 10.9091 314.1788 381.0759 Tm
( )Tj
0.5973 0.04 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 326.149 382.6032 Tm
(x)Tj
/F1 1 Tf
0.598 0 TD
(+)Tj
/F14 1 Tf
0.8265 0 TD
(h)Tj
/F9 1 Tf
10.9091 0 0 10.9091 326.386 379.3305 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 331.8814 356.4214 Tm
(x)Tj
/F8 1 Tf
10.9091 0 0 10.9091 344.2003 368.9673 Tm
()Tj
/F2 1 Tf
0.6257 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5032 0 TD
[(\))-166.6(log)]TJ
/F8 1 Tf
2.0139 0 TD
(p)Tj
/F9 1 Tf
0.5031 1.15 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F2 1 Tf
0.5555 0.05 TD
(+)Tj
/F8 1 Tf
1 0 TD
(O)Tj
/F9 1 Tf
0.7906 1.11 TD
( )Tj
0.7638 -0.16 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 423.6752 355.1711 Tm
(p)Tj
/F15 1 Tf
0.5349 0 TD
(|)Tj
/F10 1 Tf
4.9813 0 0 4.9813 431.5985 358.6909 Tm
(q)Tj
ET
431.49 357.343 3.685 -0.373 re
f
BT
4.9813 0 0 4.9813 431.486 352.0458 Tm
(d)Tj
/F2 1 Tf
10.9091 0 0 10.9091 439.7181 368.9673 Tm
(log)Tj
/F8 1 Tf
1.4584 0 TD
(p)Tj
/F9 1 Tf
0.5031 1.11 TD
-0.0001 Tc
(\012\012)Tj
/F2 1 Tf
-21.1995 -5.1586 TD
0 Tc
(=)Tj
/F9 1 Tf
0.9445 0.95 TD
()Tj
/F10 1 Tf
4.9813 0 0 4.9813 243.3166 313.2463 Tm
(d)Tj
/F11 1 Tf
0.7396 0 TD
(|)Tj
/F10 1 Tf
0.4584 0 TD
(q)Tj
-1.5034 -1.3594 TD
-0.0001 Tc
(d>)Tj
/F12 1 Tf
1.823 0 TD
0 Tc
(1)Tj
/F9 1 Tf
10.9091 0 0 10.9091 280.2541 335.1641 Tm
()Tj
/F15 1 Tf
7.9701 0 0 7.9701 296.011 333.0999 Tm
( )Tj
/F14 1 Tf
-3.4346 -2.7722 TD
(\012)Tj
/F1 1 Tf
1.7222 0 TD
0.0001 Tc
[(\(mo)-29.3(d)]TJ
/F14 1 Tf
2.8039 0 TD
0 Tc
(d)Tj
/F1 1 Tf
0.5466 0 TD
(\))Tj
/F9 1 Tf
10.9091 0 0 10.9091 314.1788 337.3459 Tm
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 319.6334 338.4369 Tm
(x)Tj
/F1 1 Tf
0.5981 0 TD
(+)Tj
/F14 1 Tf
0.8264 0 TD
(h)Tj
/F9 1 Tf
10.9091 0 0 10.9091 319.8715 335.1641 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 325.3669 312.2551 Tm
(x)Tj
/F8 1 Tf
10.9091 0 0 10.9091 337.6847 324.8 Tm
()Tj
/F2 1 Tf
0.6257 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
[(\))-166.6(log)]TJ
/F8 1 Tf
2.0139 0 TD
(p)Tj
/F9 1 Tf
0.5031 1.15 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F2 1 Tf
0.5556 0.05 TD
(+)Tj
/F8 1 Tf
1 0 TD
(O)Tj
/F2 1 Tf
0.7905 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(d)Tj
/F2 1 Tf
0.5205 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
[(\))-166.7(log)]TJ
/F8 1 Tf
2.0139 0 TD
(q)Tj
/F2 1 Tf
0.4823 0 TD
(\))Tj
/F8 1 Tf
0.389 0 TD
(.)Tj
/F2 1 Tf
-31.111 -3.4636 TD
[(Th)27.8(us)]TJ
/F9 1 Tf
0.0399 -3.2115 TD
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 118.7995 228.3527 Tm
(q)Tj
/F15 1 Tf
0.5103 0 TD
(Q)Tj
/F9 1 Tf
10.9091 0 0 10.9091 139.8625 251.9801 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 137.2463 228.264 Tm
(\012)Tj
/F15 1 Tf
0.6596 0 TD
()Tj
/F1 1 Tf
T*
(=)Tj
/F14 1 Tf
0.8265 0 TD
(\012)Tj
/F12 1 Tf
4.9813 0 0 4.9813 154.3482 227.2677 Tm
(0)Tj
/F9 1 Tf
10.9091 0 0 10.9091 160.0548 254.1619 Tm
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 165.5093 255.2528 Tm
(x)Tj
/F1 1 Tf
0.5981 0 TD
(+)Tj
/F14 1 Tf
0.8264 0 TD
(h)Tj
/F9 1 Tf
10.9091 0 0 10.9091 165.7475 251.9801 Tm
()Tj
/F14 1 Tf
7.9701 0 0 7.9701 171.2428 229.071 Tm
(x)Tj
/F8 1 Tf
10.9091 0 0 10.9091 183.5606 241.6169 Tm
()Tj
/F2 1 Tf
0.6258 0 TD
(\()Tj
/F8 1 Tf
0.3889 0 TD
(p)Tj
/F2 1 Tf
0.5031 0 TD
[(\))-166.7(log)]TJ
/F8 1 Tf
2.0139 0 TD
(p)Tj
/F9 1 Tf
0.5031 1.15 TD
()Tj
0 -0.6 TD
()Tj
T*
()Tj
/F2 1 Tf
0.6111 0.05 TD
(=)Tj
/F9 1 Tf
1.6362 0.95 TD
()Tj
/F1 1 Tf
7.9701 0 0 7.9701 245.7574 228.264 Tm
(1)Tj
/F14 1 Tf
0.5312 0 TD
(