{
  "id": "0904.1051",
  "version": "v1",
  "published": "2009-04-07T04:25:58.000Z",
  "updated": "2009-04-07T04:25:58.000Z",
  "title": "The argument of the Riemann $Ξ$-function off the critical line",
  "authors": [
    "Xiannan Li"
  ],
  "comment": "9 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We examine the behaviour of the zeros of the real and imaginary parts of $\\xi(s)$ on the vertical line $\\Re s = 1/2+\\lambda$, for $\\lambda \\neq 0$. This can be rephrased in terms of studying the zeros of families of entire functions $A(s) = {1/2} (\\xi(s+\\lambda) + \\xi(s - \\lambda))$ and $B(s) = \\frac{1}{2i} (\\xi(s+\\lambda) - \\xi(s - \\lambda))$. We will prove some unconditional analogues of results appearing in \\cite{Lag}, specifically that the normalized spacings of the zeros of these functions converges to a limiting distribution consisting of equal spacings of length 1, in contrast to the expected GUE distribution for the same zeros at $\\lambda = 0$. We will also show that, outside of a small exceptional set, the zeros of $\\Re \\xi(s)$ and $\\Im \\xi(s)$ interlace on $\\Re s = 1/2+\\lambda$. These results will depend on showing that away from the critical line, $\\arg \\xi(s)$ is well behaved.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-04-07T04:25:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26"
    ],
    "keywords": [
      "critical line",
      "small exceptional set",
      "imaginary parts",
      "expected gue distribution",
      "equal spacings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}