{
  "id": "2301.03165",
  "version": "v1",
  "published": "2023-01-09T04:42:34.000Z",
  "updated": "2023-01-09T04:42:34.000Z",
  "title": "Explicit bounds on $ζ(s)$ in the critical strip and a zero-free region",
  "authors": [
    "Andrew Yang"
  ],
  "comment": "51 pages. Comments are welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "We derive explicit upper bounds for the Riemann zeta-function $\\zeta(\\sigma + it)$ on the lines $\\sigma = 1 - k/(2^k - 2)$ for integer $k \\ge 4$. This is used to show that the zeta-function has no zeroes in the region $$\\sigma > 1 - \\frac{\\log\\log|t|}{21.432\\log|t|},\\qquad |t| \\ge 3.$$ This is the largest known zero-free region for $\\exp(209) \\le t \\le \\exp(5\\cdot 10^{5})$. Our results rely on an explicit version of the van der Corput $A^nB$ process for bounding exponential sums.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-09T04:42:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26",
      "11Y35"
    ],
    "keywords": [
      "zero-free region",
      "explicit bounds",
      "critical strip",
      "van der corput",
      "derive explicit upper bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}