# A Framework for Modular Properties of False Theta Functions

## TL;DR

This paper introduces a modular completion framework for false theta functions, enabling efficient computation of their modular obstructions and applications to number theory and quantum modular forms.

## Contribution

It provides the first systematic method to obtain modular completions for false theta functions, facilitating their analysis and application.

## Key findings

- Derived a convergent Rademacher-type exact formula for unimodal sequences
- Extended asymptotic analysis of unimodal sequences
- Rederived quantum modular properties from the new modular completion

## Abstract

False theta functions closely resemble ordinary theta functions, however they do not have the modular transformation properties that theta functions have. In this paper, we find modular completions for false theta functions, which among other things gives an efficient way to compute their obstruction to modularity. This has potential applications for a variety of contexts where false and partial theta series appear. To exemplify the utility of this derivation, we discuss the details of its use on two cases. First, we derive a convergent Rademacher-type exact formula for the number of unimodal sequences via the Circle Method and extend earlier work on their asymptotic properties. Secondly, we show how quantum modular properties of the limits of false theta functions can be rederived directly from the modular completion of false theta functions proposed in this paper.