{
  "id": "1910.08452",
  "version": "v1",
  "published": "2019-10-18T14:56:01.000Z",
  "updated": "2019-10-18T14:56:01.000Z",
  "title": "On two conjectures of Sierpiński concerning the arithmetic functions $σ$ and $φ$",
  "authors": [
    "Kevin Ford",
    "Sergei Konyagin"
  ],
  "comment": "published in 1999",
  "journal": "Number Theory in Progress (Zakopane, Poland 1997; K\\'alm\\'an Gy\\\"ory, Henryk Iwaniec, Jerzy Urbanowicz, eds), de Gruyter (1999), 795-803",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\sigma(n)$ denote the sum of the positive divisors of $n$. We prove that for any positive integer $k$, there is a number $m$ for which the equation $\\sigma(x)=m$ has exactly $k$ solutions, settling a conjecture of Sierpi\\'nski from 1955. Additionally, it is shown that for every positive even $k$, there is a number $m$ for which the equation $\\phi(x)=m$ has exactly $k$ solutions, where $\\phi$ is Euler's function, making progress toward another conjecture of Sierpi\\'nski from 1955.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-18T14:56:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11N64",
      "11N36"
    ],
    "keywords": [
      "arithmetic functions",
      "sierpiński concerning",
      "conjecture",
      "sierpinski",
      "eulers function"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}