Primes in arithmetic progressions to large moduli II: well-factorable estimates

We establish new mean value theorems for primes of size x in arithmetic progressions to moduli as large as x^3/5−ϵ when summed with suitably well-factorable weights. This extends well-known work of Bombieri, Friedlander and Iwaniec, who handled moduli of size at most x^4/7−ϵ. This has consequences for the level of distribution for sieve weights coming from the linear sieve.