Refinements to the prime number theorem for arithmetic progressions

Abstract

We prove a version of the prime number theorem for arithmetic progressions that is uniform enough to deduce the Siegel-Walfisz theorem, Hoheisel's asymptotic for intervals of length x 1−δ , a Brun-Titchmarsh bound, and Linnik's bound on the least prime in an arithmetic progression as corollaries. Our proof uses the Vinogradov-Korobov zero-free region and a refinement of Bombieri's "repulsive" log-free zero density estimate. Improvements exist when the modulus is sufficiently powerful.