{
  "id": "1901.04061",
  "version": "v1",
  "published": "2019-01-13T21:04:02.000Z",
  "updated": "2019-01-13T21:04:02.000Z",
  "title": "Moments of the Riemann zeta function on short intervals of the critical line",
  "authors": [
    "Louis-Pierre Arguin",
    "Frédéric Ouimet",
    "Maksym Radziwiłł"
  ],
  "comment": "34 pages, 1 figure",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that as $T\\to \\infty$, for all $t\\in [T,2T]$ outside of a set of measure $\\mathrm{o}(T)$, $$ \\int_{-(\\log T)^{\\theta}}^{(\\log T)^{\\theta}} |\\zeta(\\tfrac 12 + \\mathrm{i} t + \\mathrm{i} h)|^{\\beta} \\mathrm{d} h = (\\log T)^{f_{\\theta}(\\beta) + \\mathrm{o}(1)}, $$ for some explicit exponent $f_{\\theta}(\\beta)$, where $\\theta > -1$ and $\\beta > 0$. This proves an extended version of a conjecture of Fyodorov and Keating (2014). In particular, it show that, for all $\\theta > -1$, the moments exhibit a phase transition at a critical exponent $\\beta_c(\\theta)$, below which $f_\\theta(\\beta)$ is quadratic and above which $f_\\theta(\\beta)$ is linear. The form of the exponent $f_\\theta$ also differs between mesoscopic intervals ($-1<\\theta<0$) and macroscopic intervals ($\\theta>0$), a phenomenon that stems from an approximate tree structure for the correlations of zeta. We also prove that, for all $t\\in [T,2T]$ outside a set of measure $\\mathrm{o}(T)$, $$ \\max_{|h| \\leq (\\log T)^{\\theta}} |\\zeta(\\tfrac{1}{2} + \\mathrm{i} t + \\mathrm{i} h)| = (\\log T)^{m(\\theta) + \\mathrm{o}(1)}, $$ for some explicit $m(\\theta)$. This generalizes earlier results of Najnudel (2018) and Arguin et al. (2018) for $\\theta = 0$. The proofs are unconditional, except for the upper bounds when $\\theta > 3$, where the Riemann hypothesis is assumed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-13T21:04:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "60F10",
      "60G60",
      "60G70"
    ],
    "keywords": [
      "riemann zeta function",
      "short intervals",
      "critical line",
      "generalizes earlier results",
      "approximate tree structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}