{
  "id": "2106.10756",
  "version": "v1",
  "published": "2021-06-20T21:41:15.000Z",
  "updated": "2021-06-20T21:41:15.000Z",
  "title": "Sums of proper divisors follow the Erdős--Kac law",
  "authors": [
    "Paul Pollack",
    "Lee Troupe"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $s(n)=\\sum_{d\\mid n,~d<n} d$ denote the sum of the proper divisors of $n$. The second-named author proved that $\\omega(s(n))$ has normal order $\\log\\log{n}$, the analogue for $s$-values of a classical result of Hardy and Ramanujan. We establish the corresponding Erd\\H{o}s--Kac theorem: $\\omega(s(n))$ is asymptotically normally distributed with mean and variance $\\log\\log{n}$. The same method applies with $s(n)$ replaced by any of several other unconventional arithmetic functions, such as $\\beta(n):=\\sum_{p\\mid n} p$, $n-\\phi(n)$, and $n+\\tau(n)$ ($\\tau$ being the divisor function).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-20T21:41:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11N56",
      "11N64"
    ],
    "keywords": [
      "proper divisors",
      "erdős-kac law",
      "unconventional arithmetic functions",
      "normal order",
      "method applies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}