{
  "id": "math/0703340",
  "version": "v1",
  "published": "2007-03-12T13:17:25.000Z",
  "updated": "2007-03-12T13:17:25.000Z",
  "title": "A spectral interpretation of the zeros of the constant term of certain Eisenstein series",
  "authors": [
    "Werner Mueller"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT",
    "math.SP"
  ],
  "abstract": "In this paper we consider the constant term $\\phi_K(y,s)$ of the non-normalized Eisenstein series attached to $\\PSL(2,\\cO_K)$, where $K$ is either $\\Q$ or an imaginary quadratic field of class number one. The main purpose of this paper is to show that for every $a\\ge 1$ the zeros of the Dirichlet series $\\phi_K(a,s)$ admit a spectral interpretation in terms of eigenvalues of a natural self-adjoint operator $\\Delta_a$. This implies that, except for at most two real zeros, all zeros of $\\phi_K(a,s)$ are on the critical line, and all zeros are simple. For $K=\\Q$ this is due to Lagarias and Suzuki and Ki.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-03-12T13:17:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M36",
      "11M26"
    ],
    "keywords": [
      "constant term",
      "spectral interpretation",
      "imaginary quadratic field",
      "natural self-adjoint operator",
      "class number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......3340M"
    }
  }
}