{
  "id": "1901.06596",
  "version": "v1",
  "published": "2019-01-19T22:19:14.000Z",
  "updated": "2019-01-19T22:19:14.000Z",
  "title": "Constants of de Bruijn-Newman type in analytic number theory and statistical physics",
  "authors": [
    "Charles M. Newman",
    "Wei Wu"
  ],
  "comment": "A review, for Bull. A.M.S., of 100 years work on the Riemann Hypothesis (RH) and zeros of stat. physics partition functions, since P\\'{o}lya's $\\lambda$-perturbation of Riemann's xi-function with $\\lambda$ real. The de Bruijn(1950)-Newman(1976) constant $\\Lambda$ is the $\\lambda$ below which some zeros are off the critical line; RH is equivalent to $\\Lambda \\leq 0$",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP",
    "math.NT"
  ],
  "abstract": "One formulation in 1859 of the Riemann Hypothesis (RH) was that the Fourier transform $H_f(z)$ of $f$ for $ z \\in \\mathbb{C}$ has only real zeros when $f(t)$ is a specific function $\\Phi (t)$. P\\'{o}lya's 1920s approach to RH extended $H_f$ to $H_{f,\\lambda}$, the Fourier transform of $e^{\\lambda t^2} f(t)$. We review developments of this approach to RH and related ones in statistical physics where $f(t)$ is replaced by a measure $d \\rho (t)$. P\\'{o}lya's work together with 1950 and 1976 results of de Bruijn and Newman, respectively, imply the existence of a finite constant $\\Lambda_{DN} = \\Lambda_{DN} (\\Phi)$ in $(-\\infty, 1/2]$ such that $H_{\\Phi,\\lambda}$ has only real zeros if and only if $\\lambda \\geq \\Lambda_{DN}$; RH is then equivalent to $\\Lambda_{DN} \\leq 0$. Recent developments include the Rodgers and Tao proof of the 1976 conjecture that $\\Lambda_{DN} \\geq 0$ (that RH, if true, is only barely so) and the Polymath 15 project improving the $1/2$ upper bound to about $0.22$. We also present examples of $\\rho$'s with differing $H_{\\rho,\\lambda}$ and $\\Lambda_{DN} (\\rho)$ behaviors; some of these are new and based on a recent weak convergence theorem of the authors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-19T22:19:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "analytic number theory",
      "statistical physics",
      "bruijn-newman type",
      "fourier transform",
      "real zeros"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}