Summation formulas for Hurwitz class numbers and other mock modular coefficients

TL;DR

This paper derives a new summation formula for mock modular form coefficients, applying it to Hurwitz class numbers and Eisenstein series, connecting them to quadratic Dirichlet L-values.

Contribution

It introduces a mock modular analogue of a classical Bessel-sum identity, utilizing L-functions of mock Eisenstein series for the first time.

Findings

Derived a summation formula for mock modular coefficients

Connected coefficients to quadratic Dirichlet L-values

Extended classical identities to mock modular context

Abstract

We prove a formula for weighted sums of the first $n$ coefficients of mock modular forms of moderate growth and apply it to Hurwitz class numbers and coefficients of negative half integral weight Eisenstein series, which take the form of certain quadratic Dirichlet $L$-values. Our formula is a mock modular version of a Bessel-sum identity proved by Chandrasekharan and Narasimhan for Dirichlet series satisfying a functional equation. Our proof utilizes $L$-functions for mock modular Eisenstein series defined by Shankadhar and Singh.