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

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).

102 Pages; Corrected statement of Lemma 18.1 (main results unchanged)

References in corpus (1)

• Uniform Titchmarsh divisor problems

Cited by in corpus (3)

• Subconvexity for $GL(3)\times GL(2)$ twists (with an appendix by Will Sawin)
• Primes in arithmetic progressions to large moduli II: Well-factorable estimates
• Harper's beyond square-root conjecture