Radial limits of mock theta functions

TL;DR

This paper explicitly determines the behavior of weight 1/2 mock theta functions at cusps, connecting them to modular forms and addressing a question posed by Rhoades.

Contribution

It provides a general explicit analysis of Ramanujan's mock theta functions at cusps, expanding understanding of their modular properties.

Findings

Explicit formulas for mock theta functions at cusps

Connection to a large family of modular forms

Resolution of Rhoades' question

Abstract

Inspired by the original definition of mock theta functions by Ramanujan, a number of authors have considered the question of explicitly determining their behavior at the cusps. Moreover, these examples have been connected to important objects such as quantum modular forms and ranks and cranks by Folsom, Ono, and Rhoades. Here we solve the general problem of understanding Ramanujan's definition explicitly for any weight $\frac12$ mock theta function, answering a question of Rhoades. Moreover, as a side product, our results give a large, explicit family of modular forms.