{
  "id": "math/0205003",
  "version": "v1",
  "published": "2002-05-01T03:16:51.000Z",
  "updated": "2002-05-01T03:16:51.000Z",
  "title": "A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis, 2",
  "authors": [
    "Luis Baez-Duarte"
  ],
  "comment": "9 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\rho(x)=x-[x]$, $\\chi=\\chi_{(0,1)}$. In $L_2(0,\\infty)$ consider the subspace $\\B$ generated by $\\{\\rho_a|a\\geq1\\}$ where $\\rho_a(x):=\\rho(\\frac{1}{ax})$. By the Nyman-Beurling criterion the Riemann hypothesis is equivalent to the statement $\\chi\\in\\bar{\\B}$. For some time it has been conjectured, and proved in the first version of this paper, posted in arXiv:math.NT/0202141 v2, that the Riemann hypothesis is equivalent to the stronger statement that $\\chi\\in\\bar{\\Bnat}$ where $\\Bnat$ is the much smaller subspace generated by $\\{\\rho_a|a\\in\\Nat\\}$. This second version differs from the first in showing that under the Riemann hypothesis for some constant $c>0$ the distance between $\\chi$ and $-\\sum_{a=1}^n\\mu(a)e^{-c\\frac{\\log a}{\\log\\log n}}\\rho_a$ is of order $(\\log\\log n)^{-1/3}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-05-01T03:16:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "riemann hypothesis",
      "nyman-beurling criterion",
      "second version differs",
      "strengthening",
      "first version"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......5003B"
    }
  }
}