{
  "id": "1208.1131",
  "version": "v1",
  "published": "2012-08-06T11:07:49.000Z",
  "updated": "2012-08-06T11:07:49.000Z",
  "title": "The Mean Value of $L(\\tfrac{1}{2},χ)$ in the Hyperelliptic Ensemble",
  "authors": [
    "J. C. Andrade",
    "J. P. Keating"
  ],
  "comment": "22 pages, To appear in Journal of Number Theory Volume 132, Issue 12, December 2012, Pages 2793-2816",
  "doi": "10.1016/j.jnt.2012.05.017",
  "categories": [
    "math.NT"
  ],
  "abstract": "We obtain an asymptotic formula for the first moment of quadratic Dirichlet $L$--functions over function fields at the central point $s=\\tfrac{1}{2}$. Specifically, we compute the expected value of $L(\\tfrac{1}{2},\\chi)$ for an ensemble of hyperelliptic curves of genus $g$ over a fixed finite field as $g\\rightarrow\\infty$. Our approach relies on the use of the analogue of the approximate functional equation for such $L$--functions. The results presented here are the function field analogues of those obtained previously by Jutila in the number-field setting and are consistent with recent general conjectures for the moments of $L$--functions motivated by Random Matrix Theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-08-06T11:07:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G20",
      "11M50",
      "14G10"
    ],
    "keywords": [
      "mean value",
      "hyperelliptic ensemble",
      "approximate functional equation",
      "random matrix theory",
      "function field analogues"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.1131A"
    }
  }
}