19 research outputs found
Asymptotics for rank and crank moments
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.Comment: 11 page
Approximate polynomial structure in additively large sets
We show that any subset of the natural numbers with positive logarithmic Banach
density contains a set that is within a factor of two of a geometric progression,
improving the bound on a previous result of the authors. Density conditions on
subsets of the natural numbers that imply the existence of approximate powers of
arithmetic progressions are developed and explore
Higher depth quantum modular forms and plumbed -manifolds
In this paper we study new invariants attached to plumbed -manifolds that were introduced by Gukov, Pei, Putrov, and Vafa. These remarkable -series at radial limits conjecturally compute WRT invariants of the corresponding plumbed -manifold. Here we investigate the series for unimodular plumbing -graphs with six vertices. We prove that for every positive definite unimodular plumbing matrix, is a depth two quantum modular form on