Square-Free Integers as Sums of Two Squares

Abstract

Let r(n) denote the number of representations of the integer n as a sum of two squares, μ(n) the Möbius function and P(x) the error term of the Gauss circle problem. Let Q(x) := Σ_n≤x|μ(n)|r(n). In this short note we shall prove that if the estimate P(x) = O(x^θ) holds, then Q(x + y − Q(x) = Ay + O(yx^−ε/2 + x^{θ + ε}), where A is a constant. In particular this asymptotic formula is true for θ = 131/416. Our result improves Krätzel’s previous result.

This work is supported by National Natural Science Foundation of China (Grant No. 10301018).