Primes in arithmetic progressions to large moduli III: Uniform residue classes

TL;DR

This paper extends the Bombieri-Vinogradov theorem to larger moduli with uniform estimates across residue classes, advancing understanding of prime distribution in arithmetic progressions.

Contribution

It introduces new mean value theorems for primes in progressions to large moduli, with uniformity across residue classes, surpassing previous limitations.

Findings

Extended Bombieri-Vinogradov theorem to moduli of size x^{1/2+δ}

Achieved uniform estimates across all residue classes

Improved understanding of prime distribution in large moduli

Abstract

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$, extending the Bombieri-Vinogradov theorem to moduli of size $x^{1/2+\delta}$ which have conveniently sized divisors. The main feature of these estimates is that they are completely uniform with respect to the residue classes considered, unlike previous works on primes in arithmetic progressions to large moduli.