Jens Nordström: On Prime Numbers in Arithmetic Progressions

Abstract: This thesis studies the distribution of prime numbers in arithmetic progressions. Two important results are stated and proven. The first is Dirichlet's theorem on arithmetic progressions which establishes a condition under which an arithmetic progression contains infinitely many prime numbers. The second is the Prime Number Theorem for arithmetic progression, an analogue of the Prime Number Theorem, that gives an asymptotic formula for the number of primes in a given arithmetic progression. The framework underlying these results is developed through the study of Dirichlet characters and L-functions. The thesis concludes with a brief discussion on other results regarding primes in arithmetic progressions and current open problems in the area.