{
  "id": "1910.09837",
  "version": "v1",
  "published": "2019-10-22T08:53:13.000Z",
  "updated": "2019-10-22T08:53:13.000Z",
  "title": "The functional equation and zeros on the critical line of the quadrilateral zeta function",
  "authors": [
    "Takashi Nakamura"
  ],
  "comment": "16 pages. The paper \"Zeros and the functional equation of the quadrilateral zeta function '' (arXiv:1712.05169) was devided into two papers \"The functional equation and zeros on the critical line of the quadrilateral zeta function'' (this paper) and \"On zeta functions composed by the Hurwitz and periodic zeta functions'' (coming soon). The paper arXiv:1712.05169 will be withdrawn before long",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $0 < a \\le 1/2$ and define the quadrilateral zeta function by $Q(s,a) := \\zeta (s,a) + \\zeta (s,1-a) + {\\rm{Li}} (s,a) + {\\rm{Li}} (s,1-a)$, where $\\zeta (s,a)$ is the Hurwitz zeta function and ${\\rm{Li}} (s,a)$ is the periodic zeta function. In the present paper, we prove that for any $0 < a \\le 1/2$, there are positive constants $A(a)$ and $T_0(a)$ such that the numbers of zeros of $Q(s,a)$ on the line segment from $1/2$ to $1/2 +iT$ is at least $A(a) T$ whenever $T \\ge T_0(a)$. We also give some remarks on the functional equations of zeta functions related to Hamburger's and Knopp's theorems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-22T08:53:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M35",
      "11M26"
    ],
    "keywords": [
      "quadrilateral zeta function",
      "functional equation",
      "critical line",
      "hurwitz zeta function",
      "periodic zeta function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}