A Unified Partial and Mock Theta Function

TL;DR

This paper introduces a unified framework connecting partial theta functions and Ramanujan's mock theta functions, enhancing understanding of their modular properties and extending the theory of modular forms.

Contribution

It provides a novel construction that unifies partial theta functions with mock theta functions, clarifying their modularity and relationship.

Findings

Unified partial and mock theta functions within a single framework

Clarified the modular properties of mock theta functions

Extended the theory of holomorphic modular forms

Abstract

Unary theta functions have played a significant role in the theory of holomorphic modular forms and modular $L$-functions. A partial theta functions is defined analogously, but the sum is over part of the integer lattice. Such sums fail to have modular properties. We give a construction which unifies these partial theta functions with the mock theta function introduced by Ramanujan. The modularity of Ramanujan's mock theta functions has only recently been understood by the work of Sander Zwegers.