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 $x^{1-\delta }$, 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, a log-free zero density estimate, and the Deuring-Heilbronn zero repulsion phenomenon. Improvements exist when the modulus is sufficiently powerful.

11 pages. Theorems 1.1 and 2.1 improved

References in corpus (2)

• The fourth moment of Dirichlet $L$-functions along a coset and the Weyl bound
• Pretentious multiplicative functions and the prime number theorem for arithmetic progressions