The additive problem for the number of representations as a sum of two squares

TL;DR

This paper advances understanding of the asymptotic behavior of sums involving the number of representations of integers as sums of two squares, especially when the shift parameter varies with the upper limit.

Contribution

It provides an improved unconditional result on the asymptotics of sums of the form _{n x} r(n)r(n+m) for varying m.

Findings

Enhanced asymptotic formulas for sums involving r(n) and r(n+m)

Unconditional results applicable to a broader range of m

Refined estimates for the distribution of sums of two squares

Abstract

We improve a previous unconditional result about the asymptotic behavior of $\sum_{n\le x} r(n)r(n+m)$ with $r(n)$ the number of representations of $n$ as a sum of two squares when $m$ may vary with $x$.