Jean Rosenlind: The prime number theorem and Dirichlet's theorem on arithmetic progressions

Abstract: We give proofs of the two famous results in analytic number theory, the prime number theorem and Dirichlet's theorem on arithmetic progressions. We will go through the basic theory of Dirichlet series with particular focus on the Riemann zeta function. We will see that Dirichlet series are analytical functions which will allow us to use methods from complex analysis to analyze any particular Dirichlet series. Applying such methods on Dirichlet series, in particular on the Riemann zeta function, we will be able to extract information which gives us number theoretical results.