Asymptotic Distribution of the Partition Crank
Number Theory
2021-08-27 v3
Abstract
The partition crank is a statistic on partitions introduced by Freeman Dyson to explain Ramanujan's congruences. In this paper, we prove that the crank is asymptotically equidistributed modulo Q, for any odd number Q. To prove this, we obtain effective bounds on the error term from Zapata Rolon's asymptotic estimate for the crank function. We then use those bounds to prove the surjectivity and strict log-subadditivity of the crank function.
Keywords
Cite
@article{arxiv.1909.12806,
title = {Asymptotic Distribution of the Partition Crank},
author = {Asimina Hamakiotes and Aaron Kriegman and Wei-Lun Tsai},
journal= {arXiv preprint arXiv:1909.12806},
year = {2021}
}
Comments
14 pages 1) Some clarifications and corrections have been made. 2) This paper will appear in Ramanujan Journal