MAT6627, Pretentious Introduction to Analytic Number Theory, Autumn Semester 2014

(La distribution des nombres premiers)

Prof: Andrew Granville
Bureau: 6153 André Aisenstadt, Tel: 343-6583; Courriel: andrew@dms.umontreal.ca

Course Book: There is none, just my notes -- see below.

Classtimes: Monday 10h30-12h30; Wednesday 10h30-12h30, in Andre-Aisenstadt 5183.
Class dates: From Sept 3 to Dec 8, 2014
Special remarks:

Course notes

Here is all of chapter 2
Course notes (19/11/2014):

Here is up to section 5 of chapter 2

Course notes (6/11/2014):

Here is up to section 1 of chapter 2

Course notes (14/10/2014):

I do plan to revise the notes for the whole course as we progress. I will make new versions available here (with the date I posted them).

En Francais, et page web complet

Course Contents/Syllabus

In this course we will prove the prime number theorem and all of the other basic theorems of analytic number theory. (For example the prime number theorem for arithmetic progressions, Linnik's Theorem, the Polya Vinogradov Theorem, etc) Since 1859 the only coherent approach to these problems has been based on Riemann's idea connecting the distribution of prime numbers to the zeros of the Riemann zeta function -- which are the zeros of an analytic continuation. Some might argue that this is "unnatural" and ask for an approach that is less far removed from the original problems. Recently Soundararajan and I have proposed a different approach to the whole subject of analytic number theory, based on our concept of pretentiousness -- recently we have realized our dream of being able to develop the whole subject in a coherent way, without using the zeros of the Riemann zeta function. This will be the first course ever given using this approach to the subject. Topics may include

  • A discussion of the role of zeros of zeta functions in traditional analytic number theory
  • The prime number theorem in terms of mean values of multiplicative functions
  • Sieving: Heuristics and the Brun-Titchmarsh Theorem. The small sieve.
  • Smooth numbers and the tail of a sum
  • Selberg's formula in his elementary proof of the prime number theorem
  • Distance between multiplicative functions. Lower and upper bounds.
  • Dirichlet series to the right of 1
  • Halasz's Theorem
  • A pretentious proof of the prime number theorem
  • Distribution of values of multiplicative functions
  • The large sieve and the pretentious large sieve
  • Multiplicative functions in arithmetic progressions
  • Primes in arithmetic progressions: Linnik's Theorem
  • Exponential sums (Pretentious Montgomery-Vaughan)
  • Fiorilli, Barban-Davenport-Hooley, Bombieri-Vinogradov Theorems
  • Polya-Vinogradov Theorem
  • Burgess's Theorem
  • Subconvexity for L-functions
  • Quantum Unique Ergodicity
  • Gaps between primes
  • Spectra of mean values of multiplicative functions
  • Limitations to Equi-Distribution
  • The circle and divisor problems