{
  "id": "2309.03958",
  "version": "v1",
  "published": "2023-09-07T18:10:43.000Z",
  "updated": "2023-09-07T18:10:43.000Z",
  "title": "Note on the mean value of the Erdős--Hooley Delta-function",
  "authors": [
    "Régis de la Bretèche",
    "Gérald Tenenbaum"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "For integer $n\\geqslant 1$ and real $u$, let $\\Delta(n,u):=|\\{d:d\\mid n,\\,{\\rm e}^u<d\\leqslant {\\rm e}^{u+1}\\}|$. The Erd\\H{o}s--Hooley Delta-function is then defined by $\\Delta(n):=\\max_{u\\in{\\mathbb R}}\\Delta(n,u).$ We improve a recent upper bound by Koukoulopoulos and Tao by showing that $\\sum_{n\\leqslant x}\\Delta(n)\\leqslant x(\\log_2x)^{2+o(1)}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-07T18:10:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11K65"
    ],
    "keywords": [
      "mean value",
      "erdős-hooley delta-function",
      "upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}