{
  "id": "2006.15661",
  "version": "v1",
  "published": "2020-06-28T17:33:25.000Z",
  "updated": "2020-06-28T17:33:25.000Z",
  "title": "Non-vanishing for cubic $L$--functions",
  "authors": [
    "Chantal David",
    "Alexandra Florea",
    "Matilde Lalin"
  ],
  "comment": "53 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove that there is a positive proportion of $L$-functions associated to cubic characters over $\\mathbb{F}_q[T]$ that do not vanish at the critical point $s=1/2$. This is achieved by computing the first mollified moment using techniques previously developed by the authors in their work on the first moment of cubic $L$-functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Lester-Radziwill, which in turn develops further ideas from the work of Soundararajan, Harper, and Radziwill-Soundararajan. We work in the non-Kummer setting when $q \\equiv 2\\pmod{3}$, but our results could be translated into the Kummer setting when $q\\equiv 1\\pmod{3}$ as well as into the number field case (assuming the Generalized Riemann Hypothesis). Our positive proportion of non-vanishing is explicit, but extremely small, due to the fact that the implied constant in the upper bound for the mollified second moment is very large.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-28T17:33:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-vanishing",
      "positive proportion",
      "sharp upper bound",
      "number field case",
      "first moment"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}