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).
Please only print out the notes for the next couple of weeks as things may change greatly. I will keep you appraised of which chapters we will cover in the near future.
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
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
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