Asymptotic Distribution of the Partition Crank

TL;DR

This paper proves that the partition crank statistic is asymptotically evenly distributed modulo any odd number, providing effective bounds and exploring its properties like surjectivity and log-subadditivity.

Contribution

It establishes the asymptotic equidistribution of the partition crank modulo odd integers and derives effective bounds for the error term.

Findings

Crank is asymptotically equidistributed modulo odd Q

Effective bounds on error terms are obtained

Crank function is shown to be surjective and strictly log-subadditive

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.