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 $x^{3/5-\epsilon}$ when summed with suitably well-factorable weights. This extends well-known work of Bombieri, Friedlander and Iwaniec, who handled moduli of size at most $x^{4/7-\epsilon}$. This has consequences for the level of distribution for sieve weights coming from the linear sieve.