Asymptotics for rank and crank moments

TL;DR

This paper establishes precise asymptotic formulas for the moments of partition rank and crank statistics, confirming that crank moments are consistently larger than rank moments, using Hardy-Ramanujan methods.

Contribution

It proves a conjecture that crank moments exceed rank moments asymptotically, refining previous conjectures and connecting to harmonic Maass forms.

Findings

Crank moments are asymptotically larger than rank moments.

The paper provides precise asymptotic estimates for these moments.

It confirms a conjecture by Bringmann and Mahlburg.

Abstract

Moments of the partition rank and crank statistics have been studied for their connections to combinatorial objects such as Durfee symbols, as well as for their connections to harmonic Maass forms. This paper proves a conjecture due to Bringmann and Mahlburg that refined a conjecture of Garvan. Garvan's conjecture states that the moments of the crank function are always larger than the moments of the rank function, even though the moments have the same main asymptotic term. The proof uses the Hardy-Ramanujan method to provide precise asymptotic estimates for rank and crank moments and their differences.