{
  "id": "2208.10786",
  "version": "v1",
  "published": "2022-08-23T07:39:40.000Z",
  "updated": "2022-08-23T07:39:40.000Z",
  "title": "Functional equation, upper bounds and analogue of Lindelöf hypothesis for the Barnes double zeta-function",
  "authors": [
    "Takashi Miyagawa"
  ],
  "comment": "15 pages, 2 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "The functional equations of the Riemann zeta function, the Hurwitz zeta function, and the Lerch zeta function have been well known for a long time and there are great importance when studying these zeta-functions. For example, the fundamental upper bounds of these functions are obtained using functional equations. In this paper, we prove a functional equations of the Barnes double zeta-function $ \\zeta_2 (s, \\alpha ; v, w ) = \\sum_{m=0}^\\infty \\sum_{n=0}^\\infty (\\alpha+vm+wn)^{-s} $. Also, applying this functional equation and the Phragm\\'en-Lindel\\\"of convexity principle, we obtain some upper bounds for $ \\zeta_2(\\sigma + it, \\alpha ; v, w) \\ (0\\leq \\sigma \\leq 2) $ with respect to $ t $ as $ t \\rightarrow \\infty $.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-23T07:39:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "functional equation",
      "barnes double zeta-function",
      "lindelöf hypothesis",
      "riemann zeta function",
      "hurwitz zeta function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}