On the sum-of-squares function
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911091483738112 |
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| author | Iudelevich, Vitalii V. |
| author_facet | Iudelevich, Vitalii V. |
| contents | 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 $r(n+1)\neq 0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_02701 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the sum-of-squares function Iudelevich, Vitalii V. Number Theory 11M06, 11N37 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 $r(n+1)\neq 0$. |
| title | On the sum-of-squares function |
| topic | Number Theory 11M06, 11N37 |
| url | https://arxiv.org/abs/2508.02701 |