{
  "id": "1406.5462",
  "version": "v1",
  "published": "2014-06-20T17:09:50.000Z",
  "updated": "2014-06-20T17:09:50.000Z",
  "title": "Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function",
  "authors": [
    "Emanuel Carneiro",
    "Vorrapan Chandee",
    "Friedrich Littmann",
    "Micah B. Milinovich"
  ],
  "comment": "to appear in J. Reine Angew. Math",
  "categories": [
    "math.NT",
    "math.CA"
  ],
  "abstract": "Montgomery's pair correlation conjecture predicts the asymptotic behavior of the function $N(T,\\beta)$ defined to be the number of pairs $\\gamma$ and $\\gamma'$ of ordinates of nontrivial zeros of the Riemann zeta-function satisfying $0<\\gamma,\\gamma'\\leq T$ and $0 < \\gamma'-\\gamma \\leq 2\\pi \\beta/\\log T$ as $T\\to \\infty$. In this paper, assuming the Riemann hypothesis, we prove upper and lower bounds for $N(T,\\beta)$, for all $\\beta >0$, using Montgomery's formula and some extremal functions of exponential type. These functions are optimal in the sense that they majorize and minorize the characteristic function of the interval $[-\\beta, \\beta]$ in a way to minimize the $L^1\\big(\\mathbb{R}, \\big\\{1 - \\big(\\frac{\\sin \\pi x}{\\pi x}\\big)^2 \\big\\}\\,dx\\big)$-error. We give a complete solution for this extremal problem using the framework of reproducing kernel Hilbert spaces of entire functions. This extends previous work by P. X. Gallagher in 1985, where the case $\\beta \\in \\frac12 \\mathbb{N}$ was considered using non-extremal majorants and minorants.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-20T17:09:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26",
      "46E22",
      "41A30"
    ],
    "keywords": [
      "riemann zeta-function",
      "montgomerys pair correlation conjecture predicts",
      "reproducing kernel hilbert spaces",
      "asymptotic behavior",
      "non-extremal majorants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.5462C"
    }
  }
}