Moments of the number of representations as sums of two prime squares

Abstract

We prove, for every fixed integer $k\ge 4$, the correct order of magnitude for the $k$th moments of the function that counts the number of representations of an integer as sums of two prime squares. The upper bound for $k=4$ was previously known up to $\log \log \log x$, and the lower bound for $k\ge 4$ was only known conditionally on a conjectural uniform version of the Green-Tao theorem on linear equations in primes by the work of Sabuncu \cite{Sabuncu2024}. As an application of our method, we give a simpler proof of the lower bounds for the moments of the shifted prime divisor function, thereby recovering the lower-bound part of Gabdullin's recent result on a conjecture of Fan and Pomerance.