{
  "id": "1804.01445",
  "version": "v2",
  "published": "2018-04-04T14:46:34.000Z",
  "updated": "2024-09-17T02:41:18.000Z",
  "title": "Average non-vanishing of Dirichlet $L$-functions at the central point",
  "authors": [
    "Kyle Pratt"
  ],
  "comment": "22 pages. Added notice of an error in the work. Corrected proof to appear later",
  "journal": "Alg. Number Th. 13 (2019) 227-249",
  "doi": "10.2140/ant.2019.13.227",
  "categories": [
    "math.NT"
  ],
  "abstract": "The Generalized Riemann Hypothesis implies that at least 50% of the central values $L \\left( \\frac{1}{2},\\chi\\right)$ are non-vanishing as $\\chi$ ranges over primitive characters modulo $q$. We show that one may unconditionally go beyond GRH, in the sense that if one averages over primitive characters modulo $q$ and averages $q$ over an interval, then at least 50.073% of the central values are non-vanishing. The proof utilizes the mollification method with a three-piece mollifier, and relies on estimates for sums of Kloosterman sums due to Deshouillers and Iwaniec. Note: The author has been made aware of an error in this work. It seems the error can be fixed, by using a different argument, and the author will present a correction in due course.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-09-17T02:41:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "central point",
      "average non-vanishing",
      "primitive characters modulo",
      "central values",
      "generalized riemann hypothesis implies"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}