Moments of moments of primes in arithmetic progressions

TL;DR

This paper investigates the behavior of moments of primes in arithmetic progressions, establishing lower bounds unconditionally and asymptotic results under GRH and LI, revealing the distribution's nature and proposing a related conjecture.

Contribution

It provides the first unconditional Omega-results for moments of primes in arithmetic progressions and characterizes their limiting distribution under GRH and LI.

Findings

Unconditional Omega-results for weighted even moments of primes in progressions.

Lower bounds for moments established under GRH.

Asymptotic formulas for moments of the limiting distribution under GRH and LI.

Abstract

We establish unconditional $\Omega$-results for all weighted even moments of primes in arithmetic progressions. We also study the moments of these moments and establish lower bounds under GRH. Finally, under GRH and LI we prove an asymptotic for all moments of the associated limiting distribution, which in turn indicates that our unconditional and GRH results are essentially best possible. Using our probabilistic results, we formulate a conjecture on the moments with a precise associated range of validity, which we believe is also best possible. This last conjecture implies a $q$-analogue of the Montgomery-Soundararajan conjecture on the Gaussian distribution of primes in short intervals. The ideas in our proofs include a novel application of positivity in the explicit formula and the combinatorics of arrays of characters which are fixed by certain involutions.