{
  "id": "1901.02216",
  "version": "v1",
  "published": "2019-01-08T09:24:45.000Z",
  "updated": "2019-01-08T09:24:45.000Z",
  "title": "Arithmetic Subderivatives and Leibniz-Additive Functions",
  "authors": [
    "Jorma K. Merikoski",
    "Pentti Haukkanen",
    "Timo Tossavainen"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We first introduce the arithmetic subderivative of a positive integer with respect to a non-empty set of primes. This notion generalizes the concepts of the arithmetic derivative and arithmetic partial derivative. More generally, we then define that an arithmetic function $f$ is Leibniz-additive if there is a nonzero-valued and completely multiplicative function $h_f$ satisfying $f(mn)=f(m)h_f(n)+f(n)h_f(m)$ for all positive integers $m$ and $n$. We study some basic properties of such functions. For example, we present conditions when an arithmetic function is Leibniz-additive and, generalizing well-known bounds for the arithmetic derivative, establish bounds for a Leibniz-additive function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-08T09:24:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11A05"
    ],
    "keywords": [
      "leibniz-additive function",
      "arithmetic subderivative",
      "arithmetic function",
      "positive integer",
      "arithmetic derivative"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}