Congruences of Hurwitz class numbers on square classes

TL;DR

This paper proves new divisibility results for Hurwitz class numbers on square classes using holomorphic projection, establishing Ramanujan-type congruences and exploring their relation to class numbers of imaginary quadratic fields.

Contribution

It extends holomorphic projection techniques to derive novel divisibility results and Ramanujan-type congruences for Hurwitz class numbers on square classes.

Findings

Established Ramanujan-type congruences for Hurwitz class numbers.

Identified divisibility patterns shared among these congruences.

Delineated a dichotomy between class number congruences and Hurwitz class number congruences.

Abstract

We extend a holomorphic projection argument of our earlier work to prove a novel divisibility result for non-holomorphic congruences of Hurwitz class numbers. This result allows us to establish Ramanujan-type congruences for Hurwitz class numbers on square classes, where the holomorphic case parallels previous work by Radu on partition congruences. We offer two applications. The first application demonstrates common divisibility features of Ramanujan-type congruences for Hurwitz class numbers. The second application provides a dichotomy between congruences for class numbers of imaginary quadratic fields and Ramanujan-type congruences for Hurwitz class numbers.