On extreme values of $r_3(n)$ in arithmetic progressions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912141709148160 |
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| author | Filaseta, Michael Klein, Jonah Sabuncu, Cihan |
| author_facet | Filaseta, Michael Klein, Jonah Sabuncu, Cihan |
| contents | For a given integer $m$ and any residue $a \pmod{m}$ that can be written as a sum of 3 squares modulo $m$, we show the existence of infinitely many integers $n \equiv a \pmod{m}$ such that the number of representations of $n$ as a sum of three squares, $r_3(n)$, satisfies $r_3(n) \gg_m \sqrt{n} \log \log n$. Consequently, we establish that there are infinitely many integers $n \equiv a \pmod{m}$ for which the Hurwitz class number $H(n)$ also satisfies $H(n) \gg_m \sqrt{n} \log \log n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_01988 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On extreme values of $r_3(n)$ in arithmetic progressions Filaseta, Michael Klein, Jonah Sabuncu, Cihan Number Theory 11E25(Primary) 11E20, 11N37 (Secondary) For a given integer $m$ and any residue $a \pmod{m}$ that can be written as a sum of 3 squares modulo $m$, we show the existence of infinitely many integers $n \equiv a \pmod{m}$ such that the number of representations of $n$ as a sum of three squares, $r_3(n)$, satisfies $r_3(n) \gg_m \sqrt{n} \log \log n$. Consequently, we establish that there are infinitely many integers $n \equiv a \pmod{m}$ for which the Hurwitz class number $H(n)$ also satisfies $H(n) \gg_m \sqrt{n} \log \log n$. |
| title | On extreme values of $r_3(n)$ in arithmetic progressions |
| topic | Number Theory 11E25(Primary) 11E20, 11N37 (Secondary) |
| url | https://arxiv.org/abs/2412.01988 |