{
  "id": "2304.07581",
  "version": "v1",
  "published": "2023-04-15T15:31:37.000Z",
  "updated": "2023-04-15T15:31:37.000Z",
  "title": "Sharp upper bound for the sixth moment of the Riemann zeta function on the critical line",
  "authors": [
    "Thi Altenschmidt"
  ],
  "comment": "11 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "The main task of this work is to give an improvement for the upper bounds of the Laplace transform $$\\int_0^{+\\infty}\\Bigl|\\zeta\\left(\\frac{1}{2}+it\\right)\\Bigr|^{2\\beta}e^{-\\delta t}dt \\ll_{\\beta,\\varepsilon} \\frac{1}{\\delta^{\\frac{\\beta-1}{2}+\\varepsilon}}, \\quad 0 < \\delta < \\frac{\\pi}{2}, \\delta \\to 0^+, \\forall \\varepsilon > 0, \\forall \\beta \\geqslant 3.$$ In particular, this implies the desired estimation for the upper bound of the sixth moment of the Riemann zeta function on the critical line $$\\int_0^T \\Bigl|\\zeta\\left(\\frac{1}{2}+it\\right)\\Bigr|^6dt \\ll_{\\varepsilon} T^{1+\\varepsilon}, \\quad T \\to +\\infty, \\forall \\varepsilon > 0.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-15T15:31:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "riemann zeta function",
      "sharp upper bound",
      "sixth moment",
      "critical line",
      "main task"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}