Theta lifts for Lorentzian lattices and coefficients of mock theta functions

TL;DR

This paper develops methods to evaluate theta lifts for Lorentzian lattices and uses these to derive recurrences for coefficients of mock theta functions, linking lattice theory with mock modular forms.

Contribution

It introduces multiple approaches to evaluate regularized theta lifts and connects these evaluations to new recurrences for mock theta function coefficients.

Findings

Formulas for theta lift values at special points involving mock theta coefficients

Recurrences for Hurwitz class numbers, spt-function, and Ramanujan's mock theta coefficients

Comparison of evaluation methods yields new identities and relations

Abstract

We evaluate regularized theta lifts for Lorentzian lattices in three different ways. In particular, we obtain formulas for their values at special points involving coefficients of mock theta functions. By comparing the different evaluations, we derive recurrences for the coefficients of mock theta functions, such as Hurwitz class numbers, Andrews' spt-function, and Ramanujan's mock theta functions.