Asymptotics for rank and crank moments

By Kathrin Bringmann, Karl Mahlburg, Robert C. Rhoades. 2009

63 citations recorded by the source corpus.

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 of two of the authors that refined a conjecture of Garvan. Garvan's original 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 refined version provides precise asymptotic estimates for both the moments and their differences. Our proof uses the Hardy–Ramanujan Circle Method, multiple sums of Bernoulli polynomials and the theory of quasimock theta functions.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/6a91397702c14c76c5ac5cf6f11f9ab648997dbf

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

Asymptotic inequalities for positive crank and rank moments (2012, background): “MR2823873 (2012g:11184) [17] K. Bringmann, K. Mahlburg, and R. Rhoades, Asymptotics for rank and crank moments, Bull. of London Math.”
Taylor coefficients of mock-Jacobi forms and moments of partition statistics (2014, methodology): “All three authors then used an extension of this method in [12] in order to prove Garvan’s Conjecture for any fixed k and sufficiently large n.”