On a modularity conjecture of Andrews, Dixit, Schultz, and Yee for a variation of Ramanujan's $\omega(q)$

TL;DR

This paper investigates the mock modular properties of a partition function related to Ramanujan's omega function, revealing its modular completion as a derivative of harmonic Maass forms and detailing its transformation behavior.

Contribution

It provides a detailed analysis of the modular completion of ar{P}_}omega(q), showing it is a derivative of harmonic Maass forms, advancing understanding of mock modularity in partition functions.

Findings

The modular completion is not a harmonic Maass form but a derivative of such forms.

The behavior under modular transformations is precisely characterized.

The image under the Maass lowering operator is in a simpler space.

Abstract

We analyze the mock modular behavior of $\bar{P}_\omega(q)$, a partition function introduced by Andrews, Dixit, Schultz, and Yee. This function arose in a study of smallest parts functions related to classical third order mock theta functions, one of which is $\omega(q)$. We find that the modular completion of $\bar{P}_\omega(q)$ is not simply a harmonic Maass form, but is instead the derivative of a linear combination of products of various harmonic Maass forms and theta functions. We precisely describe its behavior under modular transformations and find that the image under the Maass lowering operator lies in a relatively simpler space.