Sums of Hurwitz class numbers, CM modular forms, and primes of the form $x^2+ny^2$

TL;DR

This paper links sums of Hurwitz class numbers over residue classes to coefficients of CM cusp forms, providing explicit formulas that relate to primes represented as quadratic forms.

Contribution

It establishes a connection between sums of Hurwitz class numbers and CM modular forms for composite moduli, offering explicit formulas based on prime representations.

Findings

Sums of Hurwitz class numbers can be expressed via CM cusp form coefficients for M=6 and 8.

Explicit formulas depend on the representation of primes as x^2+ny^2.

The results extend understanding of class number sums in relation to modular forms.

Abstract

We consider sums of Hurwitz class numbers of the type $\sum_{t \equiv m \pmod{M}}H(4p-t^2)$, where $M$ is composite. For $M=6$ and $8$, we show that these sums can be expressed in terms of coefficients of CM cusp forms. This leads to explicit formulas which depend on the expression of $p$ in the form $x^2+ny^2$.