A variant of the Bombieri-Vinogradov theorem with explicit constants and applications

TL;DR

This paper develops an effective version with explicit constants of a mean value theorem related to Dirichlet characters, leading to an explicit Bombieri-Vinogradov theorem with applications to shifted primes.

Contribution

It provides the first explicit constants version of a mean value theorem and the Bombieri-Vinogradov theorem, enabling precise numerical results in number theory.

Findings

Explicit constants for the mean value theorem of Vaughan.

An effective Bombieri-Vinogradov theorem with explicit bounds.

Applications to problems involving shifted primes.

Abstract

We give an effective version with explicit constants of a mean value theorem of Vaughan related to the values of \psi(y, \chi), the twisted summatory function associated to the von Mangoldt function \Lambda and a Dirichlet character \chi. As a consequence of this result we prove an effective variant of the Bombieri-Vinogradov theorem with explicit constants. This effective variant has the potential to provide explicit results in many problems. We give examples of such results in several number theoretical problems related to shifted primes.