{
  "id": "1803.06849",
  "version": "v1",
  "published": "2018-03-19T09:51:32.000Z",
  "updated": "2018-03-19T09:51:32.000Z",
  "title": "The arithmetic derivative and Leibniz-additive functions",
  "authors": [
    "Pentti Haukkanen",
    "Jorma K. Merikoski",
    "Timo Tossavainen"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "An arithmetic function $f$ is Leibniz-additive if there is a completely multiplicative function $h_f$, i.e., $h_f(1)=1$ and $h_f(mn)=h_f(m)h_f(n)$ for all positive integers $m$ and $n$, satisfying $$ f(mn)=f(m)h_f(n)+f(n)h_f(m) $$ for all positive integers $m$ and $n$. A motivation for the present study is the fact that Leibniz-additive functions are generalizations of the arithmetic derivative $D$; namely, $D$ is Leibniz-additive with $h_D(n)=n$. In this paper, we study the basic properties of Leibniz-additive functions and, among other things, show that a Leibniz-additive function $f$ is totally determined by the values of $f$ and $h_f$ at primes. We also consider properties of Leibniz-additive functions with respect to the usual product, composition and Dirichlet convolution of arithmetic functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-19T09:51:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25"
    ],
    "keywords": [
      "leibniz-additive function",
      "arithmetic derivative",
      "arithmetic function",
      "positive integers",
      "basic properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}