A variant of Bombieri-Vinogradov theorem with explicit constants

TL;DR

This paper enhances the Bombieri-Vinogradov theorem by refining Vaughan's inequality and employing Helfgott's explicit M"obius function inequality, leading to more precise bounds in prime distribution analysis.

Contribution

It introduces an improved version of Vaughan's inequality and applies Helfgott's explicit M"obius inequality to strengthen the Bombieri-Vinogradov theorem with explicit constants.

Findings

Achieved a factor of log x improvement over previous results

Provided explicit constants for the refined theorem

Enhanced understanding of prime distribution in arithmetic progressions

Abstract

In this paper we improve the result of Akbary and Hambrook by a factor of log x by obtaining a better version of Vaughan's inequality and using the explicit variant of an inequality connected to the M\"obius function, derived by Helfgott in his work on ternary Goldbach conjecture.