{
  "id": "2108.02301",
  "version": "v1",
  "published": "2021-08-04T22:17:34.000Z",
  "updated": "2021-08-04T22:17:34.000Z",
  "title": "Extreme values of derivatives of the Riemann zeta function",
  "authors": [
    "Daodao Yang"
  ],
  "comment": "27 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "It is proved that if $T$ is sufficiently large, then uniformly for all positive integers $\\ell \\leqslant (\\log T) / (\\log_2 T)$, we have \\begin{equation*} \\max_{T\\leqslant t\\leqslant 2T}\\left|\\zeta^{(\\ell)}\\Big(1+it\\Big)\\right| \\geqslant e^{\\gamma}\\cdot \\ell^{\\ell}\\cdot (\\ell+1)^{ -(\\ell+1)}\\cdot\\Big(\\log_2 T - \\log_3 T + O(1)\\Big)^{\\ell+1} \\,, \\end{equation*} where $\\gamma$ is the Euler constant. We also establish lower bounds for maximum of $\\big|\\zeta^{(\\ell)}(\\sigma+it)\\big|$ when $\\ell \\in \\mathbb N $ and $\\sigma \\in [1/2, \\,1)$ are fixed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-04T22:17:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "riemann zeta function",
      "extreme values",
      "derivatives",
      "euler constant",
      "establish lower bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}