Prime number races

Abstract

In this thesis we investigate the behaviour of primes in arithmetic progressions, with a focus on the phenomenon known as Chebyshev’s bias. Under the assumption of the Generalized Riemann Hypothesis and the Linear Independence Hypothesis, we prove that there is a bias towards quadratic non-residues. Additionally we extend the investigation to the setting of function fields. In the function field setting, we investigate the behaviour of prime polynomials in residue classes modulo a fixed monic polynomial. Moreover, we prove that for an irreducible polynomial m there is a bias towards quadratic non-residues modulo m.