On primes in arithmetic progressions and bounded gaps between many primes

We prove that the primes below $x$ are, on average, equidistributed in arithmetic progressions to smooth moduli of size up to $x^{1/2+1/40-\epsilon}$. The exponent of distribution $\tfrac{1}{2} + \tfrac{1}{40}$ improves on a result of Polymath, who had previously obtained the exponent $\tfrac{1}{2} + \tfrac{7}{300}$. As a consequence, we improve results on intervals of bounded length which contain many primes, showing that $\liminf_{n \rightarrow \infty} (p_{n+m}-p_n) = O(\exp(3.8075 m))$. The main new ingredient of our proof is a modification of the q-van der Corput process. It allows us to exploit additional averaging for the exponential sums which appear in the Type I estimates of Polymath.