{
  "id": "2007.10533",
  "version": "v1",
  "published": "2020-07-21T00:02:16.000Z",
  "updated": "2020-07-21T00:02:16.000Z",
  "title": "On The Logarithm of the Riemann zeta-function Near the Nontrivial Zeros",
  "authors": [
    "Fatma Cicek"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Assuming the Riemann hypothesis and Montgomery's Pair Correlation Conjecture, we investigate the distribution of the sequences $(\\log|\\zeta(\\rho+z)|)$ and $(\\arg\\zeta(\\rho+z)).$ Here $\\rho=\\frac12+i\\gamma$ runs over the nontrivial zeros of the zeta-function, $0<\\gamma \\leq T,$ $T$ is a large real number, and $z=u+iv$ is a nonzero complex number of modulus $\\ll 1/\\log T.$ Our approach proceeds via a study of the integral moments of these sequences. If we let $z$ tend to $0$ and further assume that all the zeros $\\rho$ are simple, we can replace the pair correlation conjecture with a weaker spacing hypothesis on the zeros and deduce that the sequence $(\\log (|\\zeta^\\prime(\\rho)|/\\log T))$ has an approximate Gaussian distribution with mean $0$ and variance $\\frac12\\log\\log T.$ This gives an alternative proof of an old result of Hejhal and improves it by providing a rate of convergence to the distribution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-21T00:02:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26"
    ],
    "keywords": [
      "nontrivial zeros",
      "riemann zeta-function",
      "montgomerys pair correlation conjecture",
      "nonzero complex number",
      "large real number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}