{
  "id": "1905.04193",
  "version": "v1",
  "published": "2019-05-10T14:27:05.000Z",
  "updated": "2019-05-10T14:27:05.000Z",
  "title": "A uniqueness property of general Dirichlet series",
  "authors": [
    "Anup B. Dixit"
  ],
  "comment": "11 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $F(s)=\\sum_n a_n/\\lambda_n^s$ be a general Dirichlet series which is absolutely convergent on $\\Re(s)>1$. Assume that $F(s)$ has an analytic continuation and satisfies a growth condition, which gives rise to certain invariants namely the degree $d_F$ and conductor $\\alpha_F$. In this paper, we show that there are at most $2d_F$ general Dirichlet series with a given degree $d_F$, conductor $\\alpha_F$ and residue $\\rho_F$ at $s=1$. As a corollary, we get that elements in the extended Selberg class with positive Dirichlet coefficients are determined by their degree, conductor and the residue at $s=1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-10T14:27:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M41"
    ],
    "keywords": [
      "general dirichlet series",
      "uniqueness property",
      "analytic continuation",
      "extended selberg class",
      "positive dirichlet coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}