Asymptotics for rank and crank moments

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.

11 pages

References in corpus (3)

• Mock Theta Functions
• Topological Strings and (Almost) Modular Forms
• Partition statistics and quasiweak Maass forms

Cited by in corpus (9)

• The first positive rank and crank moments for overpartitions
• A refinement of a result of Andrews and Newman on the sum of minimal excludants
• Higher order spt functions for overpartitions, overpartitions with smallest part even, and partitions without repeated odd parts
• Almost harmonic Maass forms and Kac-Wakimoto characters
• An inequality between finite analogues of rank and crank moments
• Asymptotic inequalities for positive crank and rank moments
• Asymtotics for Moments of Higher Ranks
• Higher Order SPT-Functions
• On Dyson's crank conjecture and the uniform asymptotic behavior of certain inverse theta functions