Simple error bounds for an asymptotic expansion of the partition function

Abstract

Recently, there has been renewed interest in studying the asymptotic properties of the integer partition function $p(n)$. Hardy, Ramanujan, and Rademacher provided detailed asymptotic analysis for $p(n)$. Presently, attention has shifted towards Poincar\'e-type asymptotic expansions, characterised by their simplicity albeit reduced accuracy compared to the earlier works of Hardy, Ramanujan, and Rademacher. This paper aims to establish computable error bounds for one such simplified expansion. The bounds presented herein are sharper, and their derivation is considerably simpler compared to those found in recent literature.