Asymptotic formulas for integer partitions within the approach of microcanonical ensemble

TL;DR

This paper applies a microcanonical ensemble approach, inspired by statistical physics, to derive asymptotic formulas for integer partitions, including corrections and extensions to two-dimensional cases.

Contribution

It introduces a novel microcanonical ensemble method to analyze integer partitions and derives asymptotic formulas with corrections for one- and two-dimensional cases.

Findings

Derived correction to the leading asymptotic for one-dimensional partitions

Obtained asymptotic estimates for two-dimensional (plane) partitions

Results align with known asymptotic formulas for plane partitions

Abstract

The problem of integer partitions is addressed using the microcanonical approach which is based on the analogy between this problem in the number theory and the calculation of microstates of a many-boson system. For ordinary (one-dimensional) partitions, the correction to the leading asymptotic is obtained. The estimate for the number of two-dimensional (plane) partitions coincides with known asymptotic results.