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arXiv · 2209.07887

Error bounds for the asymptotic expansion of the partition function

Koustav Banerje·Peter Paule·Cristian-Silviu Radu·Carsten Schneider
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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.

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Koustav Banerje, Peter Paule, Cristian-Silviu Radu, Carsten Schneider. 2022-09-16. Error bounds for the asymptotic expansion of the partition function. https://arxiv.org/abs/2209.07887

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