{
  "id": "1712.05169",
  "version": "v1",
  "published": "2017-12-14T11:01:53.000Z",
  "updated": "2017-12-14T11:01:53.000Z",
  "title": "On zeros of the bilateral Hurwitz and periodic zeta functions",
  "authors": [
    "Takashi Nakamura"
  ],
  "comment": "7 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we show that all real zeros of the bilateral Hurwitz zeta function $Z(s,a):=\\zeta (s,a) + \\zeta (s,1-a)$ with $1/4 \\le a \\le 1/2$ are on the non-positive even integers as well as the real zeros of $\\zeta (s,1/2) = (2^s-1) \\zeta (s)$. We also prove that all real zeros of the bilateral periodic zeta function $P(s,a):={\\rm{Li}}_s (e^{2\\pi ia}) + {\\rm{Li}}_s (e^{2\\pi i(1-a)})$ with $1/4 \\le a \\le 1/2$ are on the negative even integers as well as the real zeros of $\\zeta (s)$. Moreover, we show that that all real zeros of the quadrilateral zeta function $Q(s,a):=Z(s,a) + P(s,a)$ with $1/4 \\le a \\le 1/2$ are on the non-positive even integers. The complex zeros of these zeta functions are also discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-14T11:01:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "real zeros",
      "bilateral hurwitz zeta function",
      "bilateral periodic zeta function",
      "quadrilateral zeta function",
      "complex zeros"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}