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We study the moments of the function that counts the number ofrepresentations of an integer as sums of two prime squares. We refine some ofthe previous arguments and apply Selberg sieve to get an unconditional upperbound for all moments. We also prove a lower bound for all moments conditionalon some generalization of Green-Tao theorem on Linear Equations in primes. Moreprecisely, for fifth moment and onward, we get the expected order of magnitudelower and upper bounds, and provide some heuristics on this representationfunction.

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