{
  "id": "1701.06608",
  "version": "v1",
  "published": "2017-01-23T19:58:31.000Z",
  "updated": "2017-01-23T19:58:31.000Z",
  "title": "Linear correlations of the divisor function",
  "authors": [
    "Sandro Bettin"
  ],
  "comment": "49 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Motivated by arithmetic applications on the number of points in a bihomogeneous variety and on moments of Dirichlet $L$-functions, we provide analytic continuation for the series $\\mathcal A_{\\boldsymbol{a}}(s):=\\sum_{n_1,\\dots,n_k\\geq1}\\frac{d(n_1)\\cdots d(n_k)}{(n_1\\cdots n_k)^{s}}$ with the sum restricted to solutions of a non-trivial linear equation $a_1n_1+\\cdots+a_kn_k=0$. The series $\\mathcal A_{\\boldsymbol{a}}(s)$ converges absolutely for $\\Re(s)>1-\\frac1k$ and we show it can be meromorphically continued to $\\Re(s)>1-\\frac 2{k+1}$ with poles at $s=1-\\frac1{k-j}$ only, for $1\\leq j< (k-1)/2$. As an application, we obtain an asymptotic formula with power saving error term for the number of points in the variety $a_1x_1y_1+\\cdots+a_kx_ky_k=0$ in $\\mathbb P^{k-1}(\\mathbb Q)\\times \\mathbb P^{k-1}(\\mathbb Q)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-23T19:58:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M41",
      "11N37",
      "11D72"
    ],
    "keywords": [
      "divisor function",
      "linear correlations",
      "non-trivial linear equation",
      "power saving error term",
      "analytic continuation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}