Dirichlet's Theorem on primes in arithmetic progressions, and Chebyshev's Bias

Abstract

In this master thesis we will prove Dirichlet's theorem on primes in arithmetic progressions. The theorem states that there are an infinite number of primes in every primitive residue class modulo any positive number q, where q is bigger than 2. The proof involves Dirichlet characters, L-series, and the von Mangoldt function. We start with looking at some results concerning the primes such as the prime number theorem. Along with the proof of Dirichlet' theorem, we show a simpler way to derive the theorem for the cases modulo q=3,4,6, which does not rely on the heavy machinery developed later on. The reader will be presented by a formula for the number of non-trivial zeros of the Riemann zeta function, which is derived by the means of complex analysis. As an extension to Dirichlet's theorem, we will also highlight its connection with Frobenius density theorem and Chebotarëv's density theorem. This will provide an algebraic perspective on the issue. In addition, we include a discussion on the phenomenon called Chebyshev's bias. It is a phenomenon noticed by Chebyshev that there are more primes in some residue classes, than others. For example are there more primes in quadratic non-residue classes than in quadratic residue classes.