On the Moments of the Number of Representations as Sums of Two Prime Squares

TL;DR

This paper investigates the statistical properties of the number of representations of integers as sums of two prime squares, providing bounds and heuristics using advanced sieve methods and assumptions related to primes.

Contribution

It offers the first unconditional upper bounds for all moments and conditional lower bounds for higher moments of the representation count, extending previous work with new techniques.

Findings

Unconditional upper bounds for all moments of the representation function.

Conditional lower bounds for moments from the fifth onward.

Heuristics on the distribution of the number of representations.

Abstract

We study the moments of the function that counts the number of representations of an integer as sums of two prime squares. We refine some of the previous arguments and apply the Selberg sieve to get an unconditional upper bound for all moments. We also prove a lower bound for all moments conditional on some generalization of the Green-Tao theorem on linear equations in primes. More precisely, for the fifth moment and onward, we get the expected order of magnitude lower and upper bounds. In addition, we provide some heuristics on the mass function of this representation function.