A Variational Barban-Davenport-Halberstam Theorem

TL;DR

This paper develops variational forms of key theorems in analytic number theory, providing new estimates for prime differences in arithmetic progressions, enhancing understanding of prime distribution.

Contribution

It introduces variational versions of the Barban-Davenport-Halberstam Theorem and large sieve inequality, offering novel tools for prime number analysis.

Findings

Established variational forms of classical theorems

Provided new bounds for sums of prime differences

Enhanced techniques for analyzing primes in progressions

Abstract

We prove variational forms of the Barban-Davenport-Halberstam Theorem and the large sieve inequality. We apply our result to prove an estimate for the sum of the squares of prime differences, averaged over arithmetic progressions.