{
  "id": "2211.06264",
  "version": "v1",
  "published": "2022-11-11T15:19:33.000Z",
  "updated": "2022-11-11T15:19:33.000Z",
  "title": "Zeros of Dirichlet $L$-functions near the critical line",
  "authors": [
    "George Dickinson"
  ],
  "comment": "28 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove an upper bound on the density of zeros very close to the critical line of the family of Dirichlet $L$-functions of modulus $q$ at height $T$. To do this, we derive an asymptotic for the twisted second moment of Dirichlet $L$-functions uniformly in $q$ and $t$. As a second application of the asymptotic formula we prove that, for every integer $q$, at least $38.2\\%$ of zeros of the primitive Dirichlet $L$-functions of modulus $q$ lie on the critical line.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-11T15:19:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical line",
      "asymptotic formula",
      "upper bound",
      "second application",
      "twisted second moment"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}