ERROR BOUNDS FOR THE ASYMPTOTIC EXPANSION OF THE PARTITION FUNCTION

Abstract

Asymptotic study on the partition function $p(n)$ began with the work of Hardy and Ramanujan (1918). Later Rademacher (1937) obtained a convergent series for $p(n)$ and an error bound was given by Lehmer (1937). Despite having this, a full asymptotic expansion for $p(n)$ with an explicit error bound is not known. Recently O’Sullivan (2023) studied the asymptotic expansion of $p^k(n)$-partitions into $k$-th powers, initiated by Wright (1934), 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 (2010). Our error term estimation predominantly relies on applications of algorithmic methods from symbolic summation.