{
  "id": "2404.13666",
  "version": "v1",
  "published": "2024-04-21T13:58:09.000Z",
  "updated": "2024-04-21T13:58:09.000Z",
  "title": "Sums of the triple divisor function over values of some quadratic forms",
  "authors": [
    "Chenhao Du",
    "Qingfeng Sun"
  ],
  "comment": "23 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\tau_3(n)$ be the triple divisor function. It is proved that $$ \\sum_{1\\leq n_1,n_2,n_3\\leq \\sqrt{x}}\\tau_3(n_1^2+n_2^2+n_3^2)=c_1x^{\\frac{3}{2}}(\\log x)^2+ c_2x^{\\frac{3}{2}}\\log x +c_3x^{\\frac{3}{2}} +O_{\\varepsilon}(x^{\\frac{13}{10}+\\varepsilon}) $$ for some constants $c_1$, $c_2$ and $c_3$, updating a result of the second author and Zhang. Moreover, we show that $$ \\sum_{1\\leq n_1,n_2,n_3,n_4\\leq \\sqrt{x}}\\tau_3(n_1^2+n_2^2+n_3^2+n_4^2) =c_4x^{2}(\\log x)^2+c_5x^{2}\\log x+c_6x^{2} +O_{\\varepsilon}\\left(x^{\\frac{5}{3}+\\varepsilon}\\right) $$ for some constants $c_4$, $c_5$ and $c_6$, which improves the results in [6].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-21T13:58:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "triple divisor function",
      "quadratic forms",
      "second author"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}