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 $x^{1\mathrm{/}2+1\mathrm{/}40-\epsilon }$. The exponent of distribution $\frac{1}{2}+\frac{1}{40}$ improves on a result of Polymath, 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 ${\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}(p_{n+m}-p_n)=O(\exp (3.8075m))$. 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.