{
  "id": "1903.12445",
  "version": "v1",
  "published": "2019-03-29T10:53:10.000Z",
  "updated": "2019-03-29T10:53:10.000Z",
  "title": "On asymptotic behaviour of Dirichlet inverse",
  "authors": [
    "Falko Baustian",
    "Vladimir Bobkov"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f(n)$ be an arithmetic function with $f(1)\\neq0$ and let $f^{-1}(n)$ be its reciprocal with respect to the Dirichlet convolution. We study the asymptotic behaviour of $|f^{-1}(n)|$ with regard to the asymptotic behaviour of $|f(n)|$ assuming that the latter one grows or decays with at most polynomial or exponential speed. As a by-product, we obtain simple but constructive upper bounds for the number of ordered factorizations of $n$ into $k$ factors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-29T10:53:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11N37",
      "11N56"
    ],
    "keywords": [
      "asymptotic behaviour",
      "dirichlet inverse",
      "arithmetic function",
      "dirichlet convolution",
      "constructive upper bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}