{
  "id": "1807.11642",
  "version": "v1",
  "published": "2018-07-31T02:50:47.000Z",
  "updated": "2018-07-31T02:50:47.000Z",
  "title": "Extreme values for $S_n(σ,t)$ near the critical line",
  "authors": [
    "Andrés Chirre"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $S(\\sigma,t)=\\frac{1}{\\pi}\\arg\\zeta(\\sigma+it)$ be the argument of the Riemann zeta function at the point $\\sigma+it$ of the critical strip. For $n\\geq 1$ and $t>0$ we define $$ S_{n}(\\sigma,t) = \\int_0^t S_{n-1}(\\sigma,\\tau)\\,d\\tau\\, + \\delta_{n,\\sigma\\,}, $$ where $\\delta_{n,\\sigma}$ is a specific constant depending on $\\sigma$ and $n$. Let $0\\leq \\beta<1$ be a fixed real number. Assuming the Riemann hypothesis, we show lower bounds for the maximum of the function $S_n(\\sigma,t)$ on the interval $T^\\beta\\leq t \\leq T$ and near to the critical line, when $n\\equiv 1\\mod 4$. Similar estimates are obtained for $|S_n(\\sigma,t)|$ when $n\\not\\equiv 1\\mod 4$. This extends a recently results of Bondarenko and Seip for a region near the critical line. In particular we obtain some omega results for these functions on the critical line.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-31T02:50:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26",
      "11N37"
    ],
    "keywords": [
      "critical line",
      "extreme values",
      "riemann zeta function",
      "omega results",
      "similar estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}