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 $x^{1/2+1/40-ϵ}$. The exponent of distribution $\frac{1}{2}+\frac{1}{40}$ improves on a result of Polymath [13], who had previously obtained the exponent $\frac{1}{2}+\frac{7}{300}$. As a consequence, we improve results on intervals of bounded length which contain many primes, showing that$\underset{n\to \infty }{\lim \inf }(p_{n+m}-p_n)=O(\exp (3.8075m\left)\right).$ 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 [13].