{
  "id": "1701.04978",
  "version": "v1",
  "published": "2017-01-18T07:39:08.000Z",
  "updated": "2017-01-18T07:39:08.000Z",
  "title": "Note on the resonance method for the Riemann zeta function",
  "authors": [
    "Andriy Bondarenko",
    "Kristian Seip"
  ],
  "comment": "To appear in \"Tribute to Victor Havin. 50 years with Hardy spaces\", to be published as a volume in the series \"Operator Theory: Advances and Applications\", Birkh\\\"auser Verlag",
  "categories": [
    "math.NT"
  ],
  "abstract": "We improve Montgomery's $\\Omega$-results for $|\\zeta(\\sigma+it)|$ in the strip $1/2<\\sigma<1$ and give in particular lower bounds for the maximum of $|\\zeta(\\sigma+it)|$ on $\\sqrt{T}\\le t \\le T$ that are uniform in $\\sigma$. We give similar lower bounds for the maximum of $|\\sum_{n\\le x} n^{-1/2-it}|$ on intervals of length much larger than $x$. We rely on our recent work on lower bounds for maxima of $|\\zeta(1/2+it)|$ on long intervals, as well as work of Soundararajan, G\\'{a}l, and others. The paper aims at displaying and clarifying the conceptually different combinatorial arguments that show up in various parts of the proofs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-18T07:39:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11C20"
    ],
    "keywords": [
      "riemann zeta function",
      "resonance method",
      "similar lower bounds",
      "long intervals",
      "paper aims"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}