{
  "id": "2101.11542",
  "version": "v1",
  "published": "2021-01-27T16:55:54.000Z",
  "updated": "2021-01-27T16:55:54.000Z",
  "title": "On Erdős's Method for Bounding the Partition Function",
  "authors": [
    "Asaf Cohen Antonir",
    "Asaf Shapira"
  ],
  "comment": "To appear in Amer. Math. Monthly",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "For fixed $m$ and $R\\subseteq \\{0,1,\\ldots,m-1\\}$, take $A$ to be the set of positive integers congruent modulo $m$ to one of the elements of $R$, and let $p_A(n)$ be the number of ways to write $n$ as a sum of elements of $A$. Nathanson proved that $\\log p_A(n) \\leq (1+o(1)) \\pi \\sqrt{2n|R|/3m}$ using a variant of a remarkably simple method devised by Erd\\H{o}s in order to bound the partition function. In this short note we describe a simpler and shorter proof of Nathanson's bound.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-27T16:55:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "partition function",
      "erdőss method",
      "positive integers congruent modulo",
      "nathansons bound",
      "shorter proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}