{
  "id": "2209.07887",
  "version": "v1",
  "published": "2022-09-16T12:27:20.000Z",
  "updated": "2022-09-16T12:27:20.000Z",
  "title": "Error bounds for the asymptotic expansion of the partition function",
  "authors": [
    "Koustav Banerje",
    "Peter Paule",
    "Cristian-Silviu Radu",
    "Carsten Schneider"
  ],
  "categories": [
    "math.NT",
    "cs.SC",
    "math.CO"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-16T12:27:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A16",
      "11P82",
      "68W30"
    ],
    "keywords": [
      "partition function",
      "error term estimation predominantly relies",
      "explicit error bound",
      "full asymptotic expansion",
      "asymptotic study"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}