Sums of class numbers and mixed mock modular forms

TL;DR

This paper links sums of class numbers to mixed mock modular forms, providing explicit formulas and proving a conjecture for prime cases using these connections.

Contribution

It introduces a novel connection between sums of class numbers and mixed mock modular forms, leading to explicit formulas and proof of a conjecture for prime cases.

Findings

Sums of class numbers are coefficients of mixed mock modular forms.

Explicit formulas for these sums are derived for specific primes.

A conjecture of Brown et al. is proved for prime n.

Abstract

In this paper, we consider sums of class numbers of the type $\sum_{m\equiv a\pmod{p}} H(4n-m^2)$, where $p$ is an odd prime, $n\in \mathbb{N},$ and $a\in \mathbb{Z}$. By showing that these are coefficients of mixed mock modular forms, we obtain explicit formulas. Using these formulas for $p=5$ and $7$, we then prove a conjecture of Brown et al. in the case that $n=\ell$ is prime.