{
  "id": "1203.1759",
  "version": "v3",
  "published": "2012-03-08T11:52:56.000Z",
  "updated": "2021-09-22T07:29:17.000Z",
  "title": "On the behaviour of $γ\\Log p$ modulo 1",
  "authors": [
    "Olivier Ramaré"
  ],
  "comment": "Fundamental flaw in the argument",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove non-trivial lower bounds for sums of type $\\sum_{p\\sim P}g(\\gamma\\Log p)$, where $g$ is a non-negative $2\\pi$-periodical function and $\\gamma$ is a given parameter. As an application we prove that $\\zeta(1+it)^{\\pm1}\\ll\\Log\\Log (9+|t|)$ and extend the zero-free region of the Riemann zeta-function.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-03-13T10:59:50.000Z",
      "comment": "This paper has been withdrawn by the author due to a serious mistake that is being investigated",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2021-09-22T07:29:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-trivial lower bounds",
      "zero-free region",
      "riemann zeta-function",
      "application",
      "periodical function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.1759R"
    }
  }
}