{
  "id": "1703.10546",
  "version": "v1",
  "published": "2017-03-30T16:02:43.000Z",
  "updated": "2017-03-30T16:02:43.000Z",
  "title": "Multidimensional divisor function on average over values of quadratic polynomial",
  "authors": [
    "Nianhong Zhou"
  ],
  "comment": "16 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $F({\\bf x})={\\bf x}^tQ_m{\\bf x}+\\mathbf{b}^t{\\bf x}+c\\in\\mathbb{Z}[{\\bf x}]$ be a quadratic polynomial in $\\ell (\\ge 3 )$ variables ${\\bf x} =(x_{1},...,x_{\\ell})$, where $F({\\bf x})$ is positive when ${\\bf x}\\in\\mathbb{R}_{\\ge 1}^{\\ell}$, $Q_m\\in {\\rm M}_{\\ell}(\\mathbb{Z})$ is an $\\ell\\times\\ell$ matrix and its discriminant $\\det\\left(Q_m^t+Q_m\\right)\\neq 0$. It gives explicit asymptotic formulas for the following sum \\[ T_{k,F}(X)=\\sum_{{\\bf x}\\in [1,X]^{\\ell}\\cap\\mathbb{Z}^{\\ell}}\\tau_{k}\\left(F({\\bf x})\\right) \\] with the help of circle method. Here $\\tau_{k}(n)=\\#\\{(x_1,x_2,...,x_{k})\\in\\mathbb{N}^{k}: n=x_1x_2...x_{k}\\}$ with $k\\in\\mathbb{Z}_{\\ge 2}$ is the multidimensional divisor function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-30T16:02:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P55",
      "11L07",
      "11N37"
    ],
    "keywords": [
      "multidimensional divisor function",
      "quadratic polynomial",
      "explicit asymptotic formulas",
      "circle method",
      "discriminant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}