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

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1\mathrm{/}2}$, extending the Bombieri-Vinogradov theorem to moduli of size $x^{1\mathrm{/}2+\delta }$ 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.