{
  "id": "2401.00358",
  "version": "v1",
  "published": "2023-12-31T00:52:49.000Z",
  "updated": "2023-12-31T00:52:49.000Z",
  "title": "Joint distribution in residue classes of families of polynomially-defined multiplicative functions",
  "authors": [
    "Akash Singha Roy"
  ],
  "comment": "63 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the distribution of families of multiplicative functions among the coprime residue classes to moduli varying uniformly in a wide range, when the multiplicative functions can be controlled by the values of polynomials at the first few prime powers. We obtain complete uniform extensions of a criterion of Narkiewicz for families of such multiplicative functions, thus also generalizing and improving upon previous work for a single such function and establishing results that are optimal in most parameters and hypotheses. As a special case of our results, for any fixed $\\epsilon>0$, the Euler totient function $\\phi(n)$ and sum of divisors function $\\sigma(n)$ are jointly asymptotically equidistributed among the reduced residue classes to moduli $q$ coprime to $6$ varying uniformly up to $(\\log x)^{(1-\\epsilon)\\alpha(q)}$, where $\\alpha(q) = \\prod_{\\ell \\mid q} (\\ell-3)/(\\ell-1)$; furthermore, the coprimality restriction is necessary and the range of $q$ is essentially optimal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-31T00:52:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11N36",
      "11N37",
      "11N64",
      "11N69"
    ],
    "keywords": [
      "polynomially-defined multiplicative functions",
      "joint distribution",
      "coprime residue classes",
      "euler totient function",
      "complete uniform extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 63,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}