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

TL;DR

The paper proves that for certain residue classes, there are infinitely many integers where the number of representations as a sum of three squares, as well as the Hurwitz class number, grow at least as fast as a specified function of n.

Contribution

It establishes the existence of infinitely many integers in specific residue classes with large values of $r_3(n)$ and $H(n)$, revealing new growth properties in these arithmetic functions.

Findings

Infinitely many $n$ with $r_3(n) o ext{large}$ in certain residue classes.

Infinitely many $n$ with $H(n) o ext{large}$ in certain residue classes.

Growth rate of $r_3(n)$ and $H(n)$ is at least $ o rac{ ext{constant}}{m} imes ext{sqrt}(n) imes ext{log log} n$.

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