{
  "id": "1706.08552",
  "version": "v1",
  "published": "2017-06-26T18:31:09.000Z",
  "updated": "2017-06-26T18:31:09.000Z",
  "title": "A spectral interpretation of zeros of certain functions",
  "authors": [
    "Kim Klinger-Logan"
  ],
  "comment": "10 pages",
  "categories": [
    "math.NT",
    "math.FA"
  ],
  "abstract": "We prove that all the zeros of a certain family of meromorphic functions are on the critical line $\\text{Re}(s)=1/2$ and are simple (except for possibly $s=1/2$), by relating the zeros to the discrete spectrum of unbounded self-adjoint operators. For example, for $h(s)$ a meromorphic function with no zeros in $\\text{Re}(s)>1/2$ with $h(s)$ real-valued on $\\mathbb{R}$, and $\\frac{h(1-s)}{h(s)}\\ll |s|^{1-\\epsilon}$ in $\\text{Re}(s)>1/2$, the only zeros of $h(s)\\pm h(1-s)$ are on the critical line. One such instance of this result is $h(s)=\\xi_k(2s)$ the completed zeta-function of a number field $k$ or, more generally, many self-dual automorphic $L$-functions. We use spectral theory suggested by results of Lax-Phillips and ColinDeVerdi\\`ere. This simplifies ideas of W. M\\\"uller, J. Lagarias, M. Suzuki, H. Ki, O. Vel\\'asquez Casta\\~n\\'on, D. Hejhal, L. de Branges and P.R. Taylor.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-26T18:31:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectral interpretation",
      "meromorphic function",
      "critical line",
      "unbounded self-adjoint operators",
      "simplifies ideas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}