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arXiv · 2508.02701

On the sum-of-squares function

Vitalii V. Iudelevich
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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 $r(n+1)\neq 0$.

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Vitalii V. Iudelevich. 2025-07-28. On the sum-of-squares function. https://arxiv.org/abs/2508.02701

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