{
  "id": "1801.03653",
  "version": "v1",
  "published": "2018-01-11T07:52:37.000Z",
  "updated": "2018-01-11T07:52:37.000Z",
  "title": "Sums of gcd-sum functions with weights concerning the Gamma function and Bernoulli polynomials",
  "authors": [
    "Isao Kiuchi",
    "Sumaia Saad Eddin"
  ],
  "comment": "18 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we establish the following two identities involving the Gamma function and Bernoulli polynomials, namely $$ \\sum_{k\\leq x}\\frac{1}{k^s} \\sum_{j=1}^{k^s}\\log\\Gamma\\left(\\frac{j}{k^s}\\right) \\sum_{\\substack{d|k d^{s}|j}}(f*\\mu)(d)\\quad {\\rm and }\\quad \\sum_{k\\leq x}\\frac{1}{k^s}\\sum_{j=0}^{k^{s}-1} B_{m}\\sum_{\\substack{d|k d^{s}|j}} (f*\\mu)(d) $$ with any fixed integer $s> 1$ and any arithmetical function $f$. We give asymptotic formulas for the above with various multiplicative functions $f$. We also consider several formulas of Dirichlet series having coefficients $\\gcd$-sum functions with weights concerning the Gamma function and Bernoulli polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-11T07:52:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11N37",
      "11Y60"
    ],
    "keywords": [
      "bernoulli polynomials",
      "gamma function",
      "weights concerning",
      "gcd-sum functions",
      "dirichlet series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}