1. Background

This is an introductory course for analytic number theory. We will use the Riemann zeta function and the L-functions to prove results concerning distribution of prime numbers. We will also cover some more advanced topics, such as the Siegel-Walfisz theorem, the Brun-Titchmarsh inequality, etc.

2. Outline

Weeks 1 and 2: arithmetic functions and their asymptotics (Yifan Yang).

Weeks 3~5: the Riemann zeta function and its functional equation, distribution of zeros of the Riemann zeta function with application to the prime number theorem, hypotheses concerning the Riemann zeta function (Yifan Yang).

Weeks 6~8: Dirichlet characters, Dirichlet L-functions and their functional equations, Siegel zeros, primes in arithmetic progressions (Yifan Yang).

Weeks 9~10: Siegel zeros and the Siegel-Walfisz theorem (Peng-Jie Wong)

Weeks 11~13: Basic sieve theory and the Brun-Titchmarsh inequality (Peng-Jie Wong)

Weeks 14~16: Bombieri-Vinogradov theorem (Peng-Jie Wong)

3. Credit: 3