Refinements to the prime number theorem for arithmetic progressions

TL;DR

This paper advances the understanding of prime distribution in arithmetic progressions by proving a refined prime number theorem that leads to several classical results, employing zero-free regions and zero density estimates.

Contribution

It introduces a new version of the prime number theorem for arithmetic progressions with enhanced uniformity, enabling derivation of multiple key theorems as corollaries.

Findings

Derived the Siegel-Walfisz theorem as a corollary.

Established Hoheisel's asymptotic for short intervals.

Provided a Brun-Titchmarsh bound and Linnik's bound on the least prime.

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.