MAT6627: the distribution of prime numbers
Université de Montréal, Fall 2016
Instructor: Dimitris Koukoulopoulos
E-mail: koukoulo AT dms.umontreal.ca
Office: 4163 Pav. André Aisenstadt
Tuesday 10h30 - 11h30 and Friday 10h30 - 12h30 (4186 Pav. Andre Aisenstadt)
The course starts on Friday September 2. There is no course during the weeks of Monday October 24 and of Monday November 7. There will be extra hours of teaching on some Tuesdays from 11h30 to 12h30 to cover the loss during the week of November 7.
Textbook and references
I will follow my own notes that can be found here.
Some other useful references are:
You can also consult thenotes of A. J. Hildebrand.
- H. L. Montgomery and R. C. Vaughan,
Multiplicative number theory. I. Classical theory.
Cambridge Studies in Advanced Mathematics, 97. Cambridge University Press, Cambridge, 2007.
- H. Iwaniec et E. Kowalski,
Analytic number theory.
American Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI, 2004.
- G. Tenenbaum,
Introduction to analytic and probabilistic number theory.
Translated from the second French edition (1995) by C. B. Thomas. Cambridge Studies in Advanced Mathematics, 46. Cambridge University Press, Cambridge, 1995.
- H. Davenport,
Multiplicative number theory. Third edition. Revised and with a preface by Hugh L. Montgomery. Graduate Texts in Mathematics, 74. Springer-Verlag, New York, 2000.
- T. Apostol, Introduction to analytic number theory. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1976.
My notes on sieve methods.
The course will cover the following subjects:
- Arithmetic functions and techniques of estimation of their partial sums
- Elementary prime number theory
- Introduction to sieve methods
- Dirichlet series and complex-analytic methods
- The prime number theorem
- Dirichlet characters
- The prime number theorem for arithmetic progressions
- The large sieve and the Bombieri-Vinogradov theorem
- Bounded gaps between prime numbers
If there is sufficient time, we will also cover Linnik's theorem and the analytic theory of L-functions.
Your grade will be determined by your performance in three homework sets and one final project, explained below. Here is the breakdown:
Final project: oral part
Final project: written part
You may discuss the homework exercises with your peers, but you must write your solutions to them on your own. For the solution of the homework problems, you may consult only my course notes as well as the ones you have taken during the class. Lastly, you must write your solutions to the homework sets in LaTeX and send them to me by e-mail before the deadline.
Homework 1 (due 23 septembre) :
Homework 2 (due 21 octobre) :
Homework 3 (due 2 decembre) :
Your final project will be based on an topic of your choice from the list below. You must give a written presentation of this topic (5-10 pages in LaTeX, including either complete proofs of sketches, depending on the lenght) and do an oral presentation of 30 minutes. orale de 30 minutes. The written part is due December 9 (do not leave this for the last moment!) and the oral presentations will take place during the week of December 12.
- Selberg, Atle;
Note on a paper by L. G. Sathe.
J. Indian Math. Soc. (N.S.) 18, (1954). 83–87.
Linnik's theorem: chapter 7 of the course notes.
The analytic theory of L-functions: chapter 10 of the course notes.
Granville, Andrew; Soundararajan, K. Large character sums: pretentious characters and the Pólya-Vinogradov theorem. J. Amer. Math. Soc. 20 (2007), no. 2, 357–384.
Hildebrand, Adolf(1-IL); Tenenbaum, Gérald(F-NANC)
On integers free of large prime factors.
Trans. Amer. Math. Soc. 296 (1986), no. 1, 265–290.
Elementary proof of the prime number theorem:
- Selberg, Atle An elementary proof of the prime-number theorem. Ann. of Math. (2) 50, (1949). 305–313. (Reviewer: A. E. Ingham) 10.0X
- Erdos, P.
On a new method in elementary number theory which leads to an elementary proof of the prime number theorem.
Proc. Nat. Acad. Sci. U. S. A. 35, (1949). 374–384.
- Ternary Goldbach conjecture: chapter 26 of the book by Davenport cited above.
- The half-dimensional sieve :
- Iwaniec, H.
The half dimensional sieve.
Acta Arith. 29 (1976), no. 1, 69–95.
- Chapitre 14 du livre "Opera de cribro"; Friedlander, John; Iwaniec, Henryk, American Mathematical Society Colloquium Publications, 57. American Mathematical Society, Providence, RI, 2010.
- Means of general multiplicative functions: the theorems of Wirsing, Delange and Halász. See chapter III.4 of the book by Tenenbaum cited above.
- Matomaki, Kaisa; Radziwill, Maksym; Multiplicative functions in short intervals. Ann. of Math. (2) 183 (2016), no. 3, 1015–1056.
Zero-density estimates of L-functions. See chapter 10 of the book by Iwaniec-Kowalski cited above.