{
  "id": "1212.5890",
  "version": "v4",
  "published": "2012-12-24T10:22:39.000Z",
  "updated": "2013-09-27T11:51:53.000Z",
  "title": "Any non-monomial polynomial of the Riemann zeta-function has complex zeros off the critical line",
  "authors": [
    "Takashi Nakamura",
    "Łukasz Pańkowski"
  ],
  "comment": "17 pages. We changed the title and the organization",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we show that any polynomial of zeta or $L$-functions with some conditions has infinitely many complex zeros off the critical line. This general result has abundant applications. By using the main result, we prove that the zeta-functions associated to symmetric matrices treated by Ibukiyama and Saito, certain spectral zeta-functions and the Euler-Zagier multiple zeta-functions have infinitely many complex zeros off the critical line. Moreover, we show that the Lindel\\\"of hypothesis for the Riemann zeta-function is equivalent to the Lindel\\\"of hypothesis for zeta-functions mentioned above despite of the existence of the zeros off the critical line. Next we prove that the Barnes multiple zeta-functions associated to rational or transcendental parameters have infinitely many zeros off the critical line. By using this fact, we show that the Shintani multiple zeta-functions have infinitely many complex zeros under some conditions. As corollaries, we show that the Mordell multiple zeta-functions, the Euler-Zagier-Hurwitz type of multiple zeta-functions and the Witten multiple zeta-functions have infinitely many complex zeros off the critical line.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-09-27T11:51:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M26",
      "11M32"
    ],
    "keywords": [
      "critical line",
      "complex zeros",
      "riemann zeta-function",
      "non-monomial polynomial",
      "barnes multiple zeta-functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1209199,
      "adsabs": "2012arXiv1212.5890N"
    }
  }
}