{
  "id": "1301.3232",
  "version": "v1",
  "published": "2013-01-15T06:18:10.000Z",
  "updated": "2013-01-15T06:18:10.000Z",
  "title": "Gaps between zeros of $ζ(s)$ and the distribution of zeros of $ζ'(s)$",
  "authors": [
    "Maksym Radziwill"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT",
    "math.CA"
  ],
  "abstract": "We settle a conjecture of Farmer and Ki in a stronger form. Roughly speaking we show that there is a positive proportion of small gaps between consecutive zeros of the zeta-function $\\zeta(s)$ if and only if there is a positive proportion of zeros of $\\zeta'(s)$ lying very closely to the half-line. Our work has applications to the Siegel zero problem. We provide a criterion for the non-existence of the Siegel zero, solely in terms of the distribution of the zeros of $\\zeta(s)$. Finally on the Riemann Hypothesis and the Pair Correlation Conjecture we obtain near optimal bounds for the number of zeros of $\\zeta'(s)$ lying very closely to the half-line. Such bounds are relevant to a deeper understanding of Levinson's method, allowing us to place one-third of the zeros of the Riemann zeta-function on the half-line.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-15T06:18:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "distribution",
      "positive proportion",
      "pair correlation conjecture",
      "siegel zero problem",
      "riemann zeta-function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}