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

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1\mathrm{/}2}$. Our main result shows that the primes are equidistributed for a fixed residue class over all moduli of size $x^{1\mathrm{/}2+\delta }$ with a ‘convenient sized’ factor. As a consequence, the expected asymptotic holds for all but $O(\delta Q)$ moduli $q\sim Q=x^{1/2+\delta}$andwegetresultsformoduliaslargeas$x^{11/21}$. 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).