{
  "id": "2405.17981",
  "version": "v1",
  "published": "2024-05-28T09:13:12.000Z",
  "updated": "2024-05-28T09:13:12.000Z",
  "title": "Explicit formulae for the mean value of products of values of Dirichlet $L$-functions at positive integers",
  "authors": [
    "Stéphane Louboutin"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $m\\ge 1$ be a rational integer. We give an explicit formula for the mean value $$\\frac{2}{\\phi(f)}\\sum_{\\chi (-1)=(-1)^m}\\vert L(m,\\chi )\\vert^2,$$ where $\\chi$ ranges over the $\\phi (f)/2$ Dirichlet characters modulo $f>2$ with the same parity as $m$. We then adapt our proof to obtain explicit means values for products of the form $L(m_1,\\chi_1)\\cdots L(m_{n-1},\\chi_{n-1})\\overline{L(m_n,\\chi_1\\cdots\\chi_{n-1})}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-28T09:13:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mean value",
      "explicit formula",
      "positive integers",
      "dirichlet characters modulo",
      "explicit means values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}