{
  "id": "math/0610581",
  "version": "v1",
  "published": "2006-10-19T08:32:54.000Z",
  "updated": "2006-10-19T08:32:54.000Z",
  "title": "On a class of arithmetic convolutions involving arbitrary sets of integers",
  "authors": [
    "László Tóth"
  ],
  "journal": "Math. Pannonica, 13 (2002), 249-263",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $d,n$ be positive integers and $S$ be an arbitrary set of positive integers. We say that $d$ is an $S$-divisor of $n$ if $d|n$ and gcd $(d,n/d)\\in S$. Consider the $S$-convolution of arithmetical functions given by (1.1), where the sum is extended over the $S$-divisors of $n$. We determine the sets $S$ such that the $S$-convolution is associative and preserves the multiplicativity of functions, respectively, and discuss other basic properties of it. We give asymptotic formulae with error terms for the functions $\\sigma_S(n)$ and $\\tau_S(n)$, representing the sum and the number of $S$-divisors of $n$, respectively, for an arbitrary $S$. We improve the remainder terms of these formulae and find the maximal orders of $\\sigma_S(n)$ and $\\tau_S(n)$ assuming additional properties of $S$. These results generalize, unify and sharpen previous ones. We also pose some problems concerning these topics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-10-19T08:32:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11N37"
    ],
    "keywords": [
      "arbitrary set",
      "arithmetic convolutions",
      "positive integers",
      "asymptotic formulae",
      "error terms"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....10581T"
    }
  }
}