{
  "id": "1611.05562",
  "version": "v1",
  "published": "2016-11-17T04:46:05.000Z",
  "updated": "2016-11-17T04:46:05.000Z",
  "title": "On the extreme values of the Riemann zeta function on random intervals of the critical line",
  "authors": [
    "Joseph Najnudel"
  ],
  "categories": [
    "math.NT",
    "math.PR"
  ],
  "abstract": "In the present paper, we show that under the Riemann hypothesis, and for fixed $h, \\epsilon > 0$, the supremum of the real and the imaginary parts of $\\log \\zeta (1/2 + it)$ for $t \\in [UT -h, UT + h]$ are in the interval $[(1-\\epsilon) \\log \\log T, (1+ \\epsilon) \\log \\log T]$ with probability tending to $1$ when $T$ goes to infinity, if $U$ is uniformly distributed in $[0,1]$. This proves a weak version of a conjecture by Fyodorov, Hiary and Keating, which has recently been intensively studied in the setting of random matrices. We also unconditionally show that the supremum of $\\Re \\log \\zeta(1/2 + it)$ is at most $\\log \\log T + g(T)$ with probability tending to $1$, $g$ being any function tending to infinity at infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-17T04:46:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M50",
      "60G15",
      "60G50",
      "60G60",
      "60G70"
    ],
    "keywords": [
      "riemann zeta function",
      "extreme values",
      "random intervals",
      "critical line",
      "riemann hypothesis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}