{
  "id": "1611.10186",
  "version": "v1",
  "published": "2016-11-30T14:48:59.000Z",
  "updated": "2016-11-30T14:48:59.000Z",
  "title": "Arithmetic functions on average over values of quadratic polynomial",
  "authors": [
    "Nianhong Zhou"
  ],
  "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 $\\det\\left(Q_m^t+Q_m\\right)\\neq 0$. This paper uses the circle method to study the following sum \\[ T(F,g;X)=\\sum_{{\\bf x}\\in [1,X]^{\\ell}\\cap\\mathbb{Z}^{\\ell}}g\\left(F({\\bf x})\\right), \\] where $g$ is an arithmetic function which satisfies proper conditions. The present paper takes the divisor function $\\tau$ and the Mangoldt function $\\Lambda$ as examples. It gives explicit asymptotic formulas for $T(F,\\tau;X)$ and $T(F,\\Lambda;X)$ with error terms. The paper finally gives some applications for the results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-30T14:48:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P55",
      "11L07",
      "11E20"
    ],
    "keywords": [
      "quadratic polynomial",
      "arithmetic function",
      "explicit asymptotic formulas",
      "satisfies proper conditions",
      "circle method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}