{
  "id": "2205.04576",
  "version": "v1",
  "published": "2022-05-09T21:48:25.000Z",
  "updated": "2022-05-09T21:48:25.000Z",
  "title": "Towards the Generalized Riemann Hypothesis using only zeros of the Riemann zeta function",
  "authors": [
    "William D. Banks"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For any real $\\beta_0\\in[\\tfrac12,1)$, let ${\\rm GRH}[\\beta_0]$ be the assertion that for every Dirichlet character $\\chi$ and all zeros $\\rho=\\beta+i\\gamma$ of $L(s,\\chi)$, one has $\\beta\\le\\beta_0$ (in particular, ${\\rm GRH}[\\frac12]$ is the Generalized Riemann Hypothesis). In this paper, we show that the validity of ${\\rm GRH}[\\frac{9}{10}]$ depends only on certain distributional properties of the zeros of the Riemann zeta function $\\zeta(s)$. No conditions are imposed on the zeros of nonprincipal Dirichlet $L$-functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-09T21:48:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26",
      "11M20"
    ],
    "keywords": [
      "riemann zeta function",
      "generalized riemann hypothesis",
      "distributional properties",
      "nonprincipal dirichlet",
      "dirichlet character"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}