Analytic Number Theory

Abstract

The theme of this chapter is the distribution of prime numbers. We will begin by giving some statements and relatively elementary proofs, before introducing the key tool: the classical theory of functions of a complex variable, of which we will give a brief overview. The two following sections contain proofs of Dirichlet’s “theorem on arithmetic progressions” and the “prime number theorem”. Dirichlet series and in particular the Riemann zeta function play a fundamental role. We will illustrate this by additionally proving the functional equation of the zeta function and by formulating the famous Riemann hypothesis.