{
  "id": "1701.02286",
  "version": "v1",
  "published": "2017-01-09T18:16:13.000Z",
  "updated": "2017-01-09T18:16:13.000Z",
  "title": "When the number of divisors is a quadratic residue",
  "authors": [
    "Olivier Bordellès"
  ],
  "comment": "9 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $q > 2$ be a prime number and define $\\lambda_q := \\left( \\frac{\\tau}{q} \\right)$ where $\\tau(n)$ is the number of divisors of $n$ and $\\left( \\frac{\\cdot}{q} \\right)$ is the Legendre symbol. When $\\tau(n)$ is a quadratic residue modulo $q$, then $\\left( \\lambda_q \\star \\mathbf{1} \\right) (n)$ could be close to the number of divisors of $n$. This is the aim of this work to compare the mean value of the function $\\lambda_q \\star \\mathbf{1}$ to the well known average order of $\\tau$. The proof reveals that the results depend heavily on the value of $\\left( \\frac{2}{q} \\right)$. A bound for short sums in the case $q=5$ is also given, using profound results from the theory of integer points close to certain smooth curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-09T18:16:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11A25",
      "11M41"
    ],
    "keywords": [
      "quadratic residue modulo",
      "integer points close",
      "mean value",
      "prime number",
      "legendre symbol"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}