Analytic theory of L-functions

In this course we present the foundations of analytic number theory based on the introduction of Dirichlet L-functions as a crucial tool to study the distribution of prime numbers. We will notably cover:

- analytic continuation and functional equation for the Riemann zeta function and Dirichlet L-functions,

- non vanishing of zeta and L-functions in relation with the distribution of primes in various subsets of the integers,

- uniformity issues: Siegel--Walfisz and Bombieri--Vinogradov,

- (time permitting): Introduction to comparative number theory, introduction to the function field analogues of zeta and L-functions.