{
  "id": "math/0701726",
  "version": "v1",
  "published": "2007-01-25T04:51:35.000Z",
  "updated": "2007-01-25T04:51:35.000Z",
  "title": "The zeros of the derivative of the Riemann zeta function near the critical line",
  "authors": [
    "Haseo Ki"
  ],
  "comment": "19 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the horizontal distribution of zeros of $\\zeta'(s)$ which are denoted as $\\rho'=\\beta'+i\\gamma'$. We assume the Riemann hypothesis which implies $\\beta'\\geqslant1/2$ for any non-real zero $\\rho'$, equality being possible only at a multiple zero of $\\zeta(s)$. In this paper we prove that $\\liminf(\\beta'-1/2)\\log\\gamma'\\not=0$ if and only if for any $c>0$ and $s=\\sigma+it$ with $|\\sigma-1/2|<c/\\log t$ $(t\\geqslant10)$ $$ \\frac{\\zeta'}{\\zeta}(s)=\\frac{1}{s-\\rho}+O(\\log t), $$ where $\\rho=1/2+i\\gamma$ is the closest zero of $\\zeta(s)$ to $s$ and the origin. We also show that if $\\liminf(\\beta'-1/2)\\log\\gamma'\\not=0$, then for any $c>0$ and $s=\\sigma+it$ ($t\\geqslant10$), we have $$ \\log\\zeta(s)=O(\\frac{(\\log t)^{2-2\\sigma}}{\\log\\log t}) $$ uniformly for $1/2+c/\\log t\\leqslant\\sigma\\leqslant\\sigma_1<1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-01-25T04:51:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M26",
      "11M06"
    ],
    "keywords": [
      "riemann zeta function",
      "critical line",
      "horizontal distribution",
      "derivative",
      "riemann hypothesis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......1726K"
    }
  }
}