On the Gauss circle problem over smooth numbers

TL;DR

This paper investigates the asymptotic behavior of a weighted sum over smooth numbers related to the Gauss circle problem, providing new insights into the distribution of sums of two squares within certain ranges.

Contribution

It offers an asymptotic evaluation of the sum of smooth numbers weighted by representations as sums of two squares for specific ranges of x and y.

Findings

Derived asymptotic formulas for _G(x,y) in certain ranges.

Enhanced understanding of the distribution of sums of two squares among smooth numbers.

Abstract

The Gauss circle problem concerns with the evaluation of $\sum_{n \leq x}r(n)$, where $r(n)$ denotes the number of representations of $n$ as sums of two squares and $x \geq 2$. Let $\Psi_G(x,y)$ denote the sum of $y$-smooth numbers below $x$ weighted by $r(n)$. In this paper, we evaluate $\Psi_G(x,y)$ asymptotically for certain ranges of $x \geq y \geq 2$.