@

author = {Koustav Banerjee and Peter Paule and Cristian-Silviu Radu and Carsten Schneider},}

title = {{Error bounds for the asymptotic expansion of the partition function}},

language = {english},

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. },

journal = {Rocky Mt J Math },

volume = {to appear},

pages = {?--?},

isbn_issn = {ISSN: 357596},

year = {2023},

note = {arXiv:2209.07887 [math.NT]},

refereed = {yes},

keywords = {partition function, asymptotic expansion, error bounds, symbolic summation},

length = {43},

url = {https://doi.org/10.35011/risc.22-13}