On extreme values of $r_3(n)$ in arithmetic progressions

Michael Filaseta; Jonah Klein; Cihan Sabuncu

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arXiv · 2412.01988

On extreme values of $r_3(n)$ in arithmetic progressions

Michael Filaseta·Jonah Klein·Cihan Sabuncu

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Abstract

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 $H(n) \gg_m \sqrt{n} \log \log n$.

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Michael Filaseta, Jonah Klein, Cihan Sabuncu. 2024-12-02. On extreme values of $r_3(n)$ in arithmetic progressions. https://arxiv.org/abs/2412.01988

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