Bounds for Coefficients of the $f(q)$ Mock Theta Function and Applications to Partition Ranks

TL;DR

This paper establishes effective bounds for the Fourier coefficients of Ramanujan's mock theta function $f(q)$ and applies these bounds to prove conjectures related to partition rank functions.

Contribution

It provides the first explicit bounds for the coefficients of $f(q)$ and uses them to resolve conjectures on partition rank properties.

Findings

Effective bounds for $eta(n)$ coefficients of $f(q)$ are derived.

Conjectures on convexity of partition rank functions are proven.

Maximal multiplicative properties of partition ranks are established.

Abstract

We compute effective bounds for $\alpha(n)$, the Fourier coefficients of Ramunujan's mock theta function $f(q)$ utilizing a finite algebraic formula due to Bruinier and Schwagenscheidt. We then use these bounds to prove two conjectures of Hou and Jagadeesan on the convexity and maximal multiplicative properties of the even and odd partition rank counting functions.