Overpartitions and class numbers of binary quadratic forms

TL;DR

This paper explores the connection between overpartition rank differences and weak Maass forms, revealing new formulas, asymptotics, and identities related to class numbers and mock theta functions.

Contribution

It establishes a link between Zagier-Eisenstein series and weak Maass forms with implications for overpartition ranks and mock theta identities.

Findings

Derived exact formulas for overpartition rank differences

Established asymptotic behaviors and congruences

Discovered new q-series identities of mock theta type

Abstract

We show that the Zagier-Eisenstein series shares its non-holomorphic part with certain weak Maass forms whose holomorphic parts are generating functions for overpartition rank differences. This has a number of consequences, including exact formulas, asymptotics, and congruences for the rank differences as well as $q$-series identities of the mock theta type.