Title: Equidistribution in number fields

To all the people that encouraged me to study mathematics and all the people I’ve met through these years, sharing my same passion for maths. The aim of this project is to address prime number distribution in two different situations: arithmetic progressions, by studying Dirichlet’s theorem, and Galois extensions, through Chebotarev’s theorem. In the first part we deal with the distribution of primes in arithmetic progressions. To do so, in the first chapter Preliminaries we introduce the concepts of Dirichlet characters and L-functions. In the second chapter Dirichlet’s theorem we continue to present the theorem, its proof and some effective versions of it: a study of the error for taking the function that counts the primes up to a point in a given arith-metic progression approximated by a multiple of the pi function, the function that counts all primes up to a point. We also introduce the theory of prime races, which compares the evolution of these functions for different arithmetic progres