{
  "id": "2211.05819",
  "version": "v1",
  "published": "2022-11-10T19:21:50.000Z",
  "updated": "2022-11-10T19:21:50.000Z",
  "title": "An Erdős-Kac theorem for integers with dense divisors",
  "authors": [
    "Gérald Tenenbaum",
    "Andreas Weingartner"
  ],
  "comment": "28 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that for large integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the number of prime factors follows an approximate normal distribution, with mean $C \\log_2 n$ and variance $V \\log_2 n$, where $C=1/(1-e^{-\\gamma})\\approx 2.280$ and $V\\approx 0.414$. This result is then generalized in two different directions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-10T19:21:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N60",
      "11N25",
      "11N37"
    ],
    "keywords": [
      "dense divisors",
      "erdős-kac theorem",
      "approximate normal distribution",
      "prime factors",
      "arbitrary constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}