{
  "id": "1810.10876",
  "version": "v1",
  "published": "2018-10-25T13:40:00.000Z",
  "updated": "2018-10-25T13:40:00.000Z",
  "title": "An extension of a result of Erdös and Zaremba",
  "authors": [
    "Michel Weber"
  ],
  "comment": "22 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Erd\\\"os and Zaremba showed that $ \\limsup_{n\\to \\infty} \\frac{\\Phi(n)}{(\\log\\log n)^2}=e^\\g$, $\\g$ being Euler's constant, where $\\Phi(n)=\\sum_{d|n} \\frac{\\log d}{d}$. We extend this result to the function $\\Psi(n)= \\sum_{d|n} \\frac{(\\log d )(\\log\\log d)}{d}$ and some other functions. We show that $ \\limsup_{n\\to \\infty}\\, \\frac{\\Psi(n)}{(\\log\\log n)^2(\\log\\log\\log n)}\\,=\\, e^\\g$. The proof requires to develop a new approach. As an application, we prove that for any $\\eta>1$, any finite sequence of reals $\\{c_k, k\\in K\\}$, $\\sum_{k,\\ell\\in K} c_kc_\\ell \\, \\frac{\\gcd(k,\\ell)^{2}}{k\\ell} \\le C(\\eta) \\sum_{\\nu\\in K} c_\\nu^2(\\log\\log\\log \\nu)^\\eta \\Psi(\\nu) $, where $C(\\eta)$ depends on $\\eta$ only. This improves a recent result obtained by the author.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-25T13:40:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D57",
      "11A05",
      "11A25"
    ],
    "keywords": [
      "finite sequence",
      "eulers constant",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}