L-series and Dirichlet’s prime number theorem

The main purpose of this chapter is a proof of the following Theorem 4.1.1, called Dirichlet’s theorem on primes in arithmetic progressions or Dirichlet’s prime number theorem.

Theorem 4.1.1 (Dirichlet’s prime number theorem). Let m and k be positive integers. If (m,k) = 1, then the arithmetic progression k +mN contains infinitely many primes [ equivalently, there are infinitely many primes p ≡ k mod m ].