Dyson's Ranks and Appell-Lerch Sums

TL;DR

This paper develops a general formula for Dyson's ranks using Appell-Lerch sums, decomposing them into modular and mock modular parts, and connects these to known results on ranks and Maass forms.

Contribution

It introduces a unified approach to analyze Dyson's ranks through decomposition into modular components, extending previous results and applying to variations of Dyson's ranks.

Findings

Derived a general formula for Dyson's ranks.

Decomposed rank generating functions into modular and mock modular parts.

Unified proof of existing results on ranks and Maass forms.

Abstract

Denote by $p(n)$ the number of partitions of $n$ and by $N(a,M;n)$ the number of partitions of $n$ with rank congruent to $a$ modulo $M$. We find and prove a general formula for Dyson's ranks by considering the deviation of the ranks from the average: \begin{equation*} D(a,M) := \sum_{n= 0}^{\infty}\left(N(a,M;n) - \frac{p(n)}{M}\right) q^n. \end{equation*} Using Appell--Lerch sum properties we decompose $D(a,M)$ into modular and mock modular parts so that the mock modular component is supported on certain arithmetic progressions, whose modulus we can control. Using our decomposition, we show how our formula gives as a straightforward consequence Atkin and Swinnerton-Dyer's results on ranks as well as Bringmann, Ono, and Rhoades's results on Maass forms. We also apply our techniques to a variation of Dyson's ranks due to Berkovitch and Garvan.