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

Abstract

For a given integer $m$ and any residue $a(\mathrm{mod}m)$ that can be written as a sum of 3 squares modulo $m$, we show the existence of infinitely many integers $n\equiv a(\mathrm{mod}m)$ 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 $n\equiv a(\mathrm{mod}m)$ for which the Hurwitz class number $H(n)$ also satisfies $H(n){\gg }_m\sqrt{n}\log \log n$.