Extreme values of $r_3(n)$ in arithmetic progressions

Abstract: A classical result of Chowla shows that the representation function $r_3(n)$, which counts the number of ways $n$ can be expressed as a sum of three squares, satisfies $$r_3(n) \gg \sqrt{n} \log\log n $$ for infinitely many integers $n$. This lower bound, in turn, also implies that $ L(1, \chi_D) \gg \log\log |D|$ holds for infinitely many fundamental discriminants $D<0$. In this talk, we will investigate whether such extremal behavior of $r_3(n)$ persists when $n$ is restricted to lie in an arithmetic progression $n\equiv a \pmod q$. \\This is joint work with Jonah Klein and Michael Filaseta.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)