Primes in arithmetic progressions to large moduli I: Fixed residue classes

TL;DR

This paper establishes new mean value theorems for primes in arithmetic progressions with large moduli, extending distribution results to moduli as large as $x^{11/21}$ using advanced analytic and algebraic techniques.

Contribution

It introduces novel mean value theorems for primes in arithmetic progressions to large moduli, combining amplification, spectral theory, and algebraic geometry methods.

Findings

Primes are equidistributed for fixed residue classes over moduli up to $x^{1/2+ ext{delta}}$

Asymptotic formulas hold for most moduli around $x^{1/2+ ext{delta}}$

Distribution results extend to moduli as large as $x^{11/21}$

Abstract

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$. Our main result shows that the primes are equidistributed for a fixed residue class over all moduli of size $x^{1/2+\delta}$ with a 'convenient sized' factor. As a consequence, the expected asymptotic holds for all but $O(\delta Q)$ moduli $q\sim Q=x^{1/2+\delta}$ and we get results for moduli as large as $x^{11/21}$. Our proof extends previous techniques of Bombieri, Fouvry, Friedlander and Iwaniec by incorporating new ideas inspired by amplification methods. We combine these with techniques of Zhang and Polymath tailored to our application. In particular, we ultimately rely on exponential sum bounds coming from the spectral theory of automorphic forms (the Kuznetsov trace formula) or from algebraic geometry (Weil and Deligne style estimates).