{
  "id": "2105.12850",
  "version": "v1",
  "published": "2021-05-26T21:19:04.000Z",
  "updated": "2021-05-26T21:19:04.000Z",
  "title": "Distribution mod $p$ of Euler's totient and the sum of proper divisors",
  "authors": [
    "Noah Lebowitz-Lockard",
    "Paul Pollack",
    "Akash Singha Roy"
  ],
  "comment": "24 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We consider the distribution in residue classes modulo primes $p$ of Euler's totient function $\\phi(n)$ and the sum-of-proper-divisors function $s(n):=\\sigma(n)-n$. We prove that the values $\\phi(n)$, for $n\\le x$, that are coprime to $p$ are asymptotically uniformly distributed among the $p-1$ coprime residue classes modulo $p$, uniformly for $5 \\le p \\le (\\log{x})^A$ (with $A$ fixed but arbitrary). We also show that the values of $s(n)$, for $n$ composite, are uniformly distributed among all $p$ residue classes modulo every $p\\le (\\log{x})^A$. These appear to be the first results of their kind where the modulus is allowed to grow substantially with $x$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-26T21:19:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11N36",
      "11N64"
    ],
    "keywords": [
      "distribution mod",
      "proper divisors",
      "residue classes modulo primes",
      "coprime residue classes modulo",
      "eulers totient function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}