On the sum-of-squares function

TL;DR

This paper derives an asymptotic formula for a sum involving the ratio of the number of representations of integers as sums of two squares, revealing new insights into the distribution of such representations.

Contribution

The paper presents a novel asymptotic formula for the sum of ratios of representations of consecutive integers as sums of two squares, expanding understanding of their distribution.

Findings

Asymptotic formula for the sum of r(n)/r(n+1) as x approaches infinity

Identification of the constant c in the asymptotic expression

Demonstration of the growth rate involving (ln x)^{-3/4}

Abstract

In this paper, we derive the following asymptotic formula $$ \mathop{{\sum}'}_{n\leqslant x}\dfrac{r(n)}{r(n+1)} = {x}{(\ln x)^{-3/4}}(c+o(1)),\ \ x \to +\infty,$$ where $r(n)$ is the number of representations of $n$ as a sum of two squares, $c$ is a positive constant, and the prime indicates summation over those $n$ for which $r(n+1)\neq 0$.