Error bounds for the asymptotic expansion of the partition function

TL;DR

This paper provides a detailed analysis of the error bounds in the asymptotic expansion of the partition function p(n), offering new inequalities and answering a longstanding question in the field.

Contribution

It introduces a comprehensive method for estimating error terms in the asymptotic expansion of p(n), extending previous work with explicit bounds and inequalities.

Findings

Derived explicit error bounds for the asymptotic expansion of p(n)

Established an infinite family of inequalities for p(n)

Answered a question posed by Chen regarding error estimation

Abstract

Asymptotic study on the partition function $p(n)$ began with the work of Hardy and Ramanujan. Later Rademacher obtained a convergent series for $p(n)$ and an error bound was given by Lehmer. Despite having this, a full asymptotic expansion for $p(n)$ with an explicit error bound is not known. Recently O'Sullivan studied the asymptotic expansion of $p^{k}(n)$-partitions into $k$th powers, initiated by Wright, and consequently obtained an asymptotic expansion for $p(n)$ along with a concise description of the coefficients involved in the expansion but without any estimation of the error term. Here we consider a detailed and comprehensive analysis on an estimation of the error term obtained by truncating the asymptotic expansion for $p(n)$ at any positive integer $n$. This gives rise to an infinite family of inequalities for $p(n)$ which finally answers to a question proposed by Chen. Our error term estimation predominantly relies on applications of algorithmic methods from symbolic summation.