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

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.