Number theory I.b - Analytic number theory

This course provides an introduction to analytic number theory. After introducing the basic definitions and methods, our aim will be to prove Dirichlet's theorem on primes in arithmetic progressions and the prime number theorem.

Covered topics include:

1. Arithmetic functions: Multiplicative functions, Dirichlet convolutions
2. Asymptotic estimates: Euler's summation formula, Summation by parts, Dirichlet's hyperbola method
3. Elementary results on the distribution of prime numbers: Chebyshev's theorem, Mertens' theorems
4. Dirichlet series: Euler product, Perron's formula
5. Primes in arithmetic progressions: Dirichlet characters, Dirichlet L-functions, Proof of Dirichlet's theorem on primes in arithmetic progressions
6. The Riemann zeta function: Analytic continuation, Functional equation, Hadamard product
7. The prime number theorem: Explicit formula, Zero-free region, Proof of the prime number theorem
8. Primes in arithmetic progressions refined: Siegel zeros, Siegel-Walfisz theorem