Primes in arithmetic progressions on average II

Abstract: A deep conjecture of Montgomery and Soundararajan on the distribution of prime numbers in short intervals of length $h$ says that the third moment is bounded by $\ll h^{\frac {3}{2}-c}$ for some $c>0$. There is in the literature some conditional evidence towards this conjecture whilst in the first article to this series we gave the first instance of unconditional evidence in the form of a bound corresponding to $\ll h^{7/5+o(1)}$. In this article we push the exponent down to $\ll h^{1+o(1)}$ which more or less is expected to be best possible.