Primes in arithmetic progressions on average II

TL;DR

This paper advances the understanding of prime distribution in short intervals by providing an improved unconditional bound on the third moment, moving closer to the conjectured optimal estimate.

Contribution

It improves the unconditional bound on the third moment of primes in short intervals from rac{7}{5}+o(1) to rac{1}{1}+o(1), approaching the conjectured optimal bound.

Findings

Bound on the third moment is rac{1}{1}+o(1) for short intervals.

Provides unconditional evidence supporting conjectures on prime distribution.

Progresses towards the optimal bound conjectured by Montgomery and Soundararajan.

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.