For a positive integer n, let r₂(n) be the number of representations of n as sums of two squares (of integers), where the convention is that different signs and different orders of the summands yield distinct representations. A famous result of Gauss shows that R(x), the sum of r₂(n) for n ≤ x, is comparable to πx. Let P(n) denote the largest prime factor of n and let S(x,y):={n ≤ x: P(n) ≤ y}. In this paper, we study the asymptotic behavior of R(x,y), the sum of r₂(n) for n in S(x,y), for various ranges of 2 ≤ y ≤ x. For y in a certain large range, we show that R(x,y) is comparable to ρ(α)πx where ρ(α) is the Dickman function and α=log x/log y. We also obtain the asymptotic behavior of the partial sum of a generalized representation function following a method of Selberg.

I thank the anonymous referee for their meticulous feedback, which improved the exposition and fixed several mathematical and typographical errors. In particular, I appreciate his pointing out the error in the proof of Theorem 2.3 in the earlier version, and suggesting a direct application of Perron's formula. Finally, I thank Krishna Alladi, George Andrews, Bruce Berndt and Atul Dixit for their feedback on the manuscript.