{
  "id": "2303.14600",
  "version": "v1",
  "published": "2023-03-26T01:29:16.000Z",
  "updated": "2023-03-26T01:29:16.000Z",
  "title": "Distribution in coprime residue classes of polynomially-defined multiplicative functions",
  "authors": [
    "Paul Pollack",
    "Akash Singha Roy"
  ],
  "comment": "post-publication version; proof of absolute irreducibility in section 6 corrected",
  "journal": "Math. Z. 303, article number 93 (2023)",
  "doi": "10.1007/s00209-023-03240-7",
  "categories": [
    "math.NT"
  ],
  "abstract": "An integer-valued multiplicative function $f$ is said to be polynomially-defined if there is a nonconstant separable polynomial $F(T)\\in \\mathbb{Z}[T]$ with $f(p)=F(p)$ for all primes $p$. We study the distribution in coprime residue classes of polynomially-defined multiplicative functions, establishing equidistribution results allowing a wide range of uniformity in the modulus $q$. For example, we show that the values $\\phi(n)$, sampled over integers $n \\le x$ with $\\phi(n)$ coprime to $q$, are asymptotically equidistributed among the coprime classes modulo $q$, uniformly for moduli $q$ coprime to $6$ that are bounded by a fixed power of $\\log{x}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-26T01:29:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11N36",
      "11N64"
    ],
    "keywords": [
      "coprime residue classes",
      "polynomially-defined multiplicative functions",
      "coprime classes modulo",
      "nonconstant separable polynomial",
      "establishing equidistribution results"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}