  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).
