{
  "id": "2410.11605",
  "version": "v1",
  "published": "2024-10-15T13:41:15.000Z",
  "updated": "2024-10-15T13:41:15.000Z",
  "title": "A variant of the Linnik-Sprindzuk theorem for simple zeros of Dirichlet L-functions",
  "authors": [
    "William D. Banks"
  ],
  "comment": "18 pages; comments are welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a primitive Dirichlet character $X$, a new hypothesis $RH_{sim}^\\dagger[X]$ is introduced, which asserts that (1) all simple zeros of $L(s,X)$ in the critical strip are located on the critical line, and (2) these zeros satisfy some specific conditions on their vertical distribution. Hypothesis $RH_{sim}^\\dagger[X]$ is likely to be true since it is a consequence of the generalized Riemann hypothesis. Assuming only the generalized Lindel\\\"of hypothesis, we show that if $RH_{sim}^\\dagger[X]$ holds for one primitive character $X$, then it holds for every such character. If this occurs, then for every character $\\chi$ (primitive or not), all simple zeros of $L(s,\\chi)$ in the critical strip are located on the critical line. In particular, Siegel zeros cannot exist in this situation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-15T13:41:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26",
      "11M20"
    ],
    "keywords": [
      "simple zeros",
      "linnik-sprindzuk theorem",
      "dirichlet l-functions",
      "critical line",
      "critical strip"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}