{
  "id": "1308.5116",
  "version": "v1",
  "published": "2013-08-23T12:45:49.000Z",
  "updated": "2013-08-23T12:45:49.000Z",
  "title": "On the distribution of the zeros of the derivative of the Riemann zeta-function",
  "authors": [
    "S. J. Lester"
  ],
  "comment": "18 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We establish an unconditional asymptotic formula describing the horizontal distribution of the zeros of the derivative of the Riemann zeta-function. For $\\Re(s)=\\sigma$ satisfying $(\\log T)^{-1/3+\\epsilon} \\leq (2\\sigma-1) \\leq (\\log \\log T)^{-2}$, we show that the number of zeros of $\\zeta'(s)$ with imaginary part between zero and $T$ and real part larger than $\\sigma$ is asymptotic to $T/(2\\pi(\\sigma-1/2))$ as $T \\rightarrow \\infty$. This agrees with a prediction from random matrix theory due to Mezzadri. Hence, for $\\sigma$ in this range the zeros of $\\zeta'(s)$ are horizontally distributed like the zeros of the derivative of characteristic polynomials of random unitary matrices are radially distributed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-08-23T12:45:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26",
      "11M50"
    ],
    "keywords": [
      "riemann zeta-function",
      "derivative",
      "random unitary matrices",
      "random matrix theory",
      "real part larger"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1017/S0305004114000413",
      "journal": "Mathematical Proceedings of the Cambridge Philosophical Society",
      "year": 2014,
      "month": "Nov",
      "volume": 157,
      "number": 3,
      "pages": 425
    },
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014MPCPS.157..425L"
    }
  }
}