Precision Asymptotics for Partitions Featuring False-Indefinite Theta Functions

TL;DR

This paper develops methods to analyze the asymptotic behavior of Fourier coefficients of false-indefinite theta functions related to Maass forms, enabling detailed asymptotic expansions and exact formulas for partition functions.

Contribution

It introduces new techniques leveraging modular properties to study asymptotics of false-indefinite theta functions associated with Maass forms.

Findings

Derived detailed asymptotic expansions for specific false-indefinite theta functions.

Established Hardy-Ramanujan-Rademacher type exact formulas under certain conditions.

Applied methods to partitions separated by parity to demonstrate the approach.

Abstract

Andrews-Dyson-Hickerson, Cohen build a striking relation between q-hypergeometric series, real quadratic fields, and Maass forms. Thanks to the works of Lewis-Zagier and Zwegers we have a complete understanding on the part of these relations pertaining to Maass forms and false-indefinite theta functions. In particular, we can systematically distinguish and study the class of false-indefinite theta functions related to Maass forms. A crucial component here is the framework of mock Maass theta functions built by Zwegers in analogy with his earlier work on indefinite theta functions and their application to Ramanujan's mock theta functions. Given this understanding, a natural question is to what extent one can utilize modular properties to investigate the asymptotic behavior of the associated Fourier coefficients, especially in view of their relevance to combinatorial objects. In this paper, we develop the relevant methods to study such a question and show that quite detailed results can be obtained on the asymptotic development, which also enable Hardy-Ramanujan-Rademacher type exact formulas under the right conditions. We develop these techniques by concentrating on a concrete example involving partitions with parts separated by parity and derive an asymptotic expansion that includes all the exponentially growing terms.