On a conjecture of Montgomery and Soundararajan

TL;DR

This paper proves lower bounds for weighted moments of primes in intervals, supporting a conjecture by Montgomery and Soundararajan, with results valid unconditionally for certain values and under the Riemann Hypothesis for all.

Contribution

It establishes lower bounds for all weighted even moments of primes, advancing understanding of prime distribution in line with Montgomery and Soundararajan's conjecture.

Findings

Unconditional lower bounds for prime moments on an unbounded set of X

Lower bounds valid for all X assuming the Riemann Hypothesis

New unconditional Omega-results for the prime counting function

Abstract

We establish lower bounds for all weighted even moments of primes up to $X$ in intervals which are in agreement with a conjecture of Montgomery and Soundararajan. Our bounds hold unconditionally for an unbounded set of values of $X$, and hold for all $X$ under the Riemann Hypothesis. We also deduce new unconditional $\Omega$-results for the classical prime counting function.