{
  "id": "1704.06158",
  "version": "v1",
  "published": "2017-04-20T14:12:53.000Z",
  "updated": "2017-04-20T14:12:53.000Z",
  "title": "Extreme values of the Riemann zeta function and its argument",
  "authors": [
    "Andriy Bondarenko",
    "Kristian Seip"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We combine our version of the resonance method with certain convolution formulas for $\\zeta(s)$ and $\\log\\, \\zeta(s)$. This leads to a new $\\Omega$ result for $|\\zeta(1/2+it)|$: The maximum of $|\\zeta(1/2+it)|$ on the interval $1 \\le t \\le T$ is at least $\\exp\\left((1+o(1)) \\sqrt{\\log T \\log\\log\\log T/\\log\\log T}\\right)$. We also obtain conditional results for $S(t):=1/\\pi$ times the argument of $\\zeta(1/2+it)$ and $S_1(t):=\\int_0^t S(\\tau)d\\tau$. On the Riemann hypothesis, the maximum of $|S(t)|$ is at least $c \\sqrt{\\log T \\log\\log\\log T/\\log\\log T}$ and the maximum of $S_1(t)$ is at least $c_1 \\sqrt{\\log T \\log\\log\\log T/(\\log\\log T)^3}$ on the interval $T^{\\beta} \\le t \\le T$ whenever $0\\le \\beta < 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-20T14:12:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "riemann zeta function",
      "extreme values",
      "conditional results",
      "resonance method",
      "riemann hypothesis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}