{
  "id": "1004.1520",
  "version": "v1",
  "published": "2010-04-09T10:43:38.000Z",
  "updated": "2010-04-09T10:43:38.000Z",
  "title": "Toy models for D. H. Lehmer's conjecture II",
  "authors": [
    "Eiichi Bannai",
    "Tsuyoshi Miezaki"
  ],
  "comment": "32 pages, 6 tables",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "In the previous paper, we studied the \"Toy models for D. H. Lehmer's conjecture\". Namely, we showed that the m-th Fourier coefficient of the weighted theta series of the $\\mathbb{Z}^2$-lattice and the $A_{2}$-lattice does not vanish, when the shell of norm $m$ of those lattices is not the empty set. In other words, the spherical 4 (resp. 6)-design does not exist among the nonempty shells in the $\\mathbb{Z}^2$-lattice (resp. $A_{2}$-lattice). This paper is the sequel to the previous paper. We take 2-dimensional lattices associated to the algebraic integers of imaginary quadratic fields whose class number is either 1 or 2, except for $\\mathbb{Q}(\\sqrt{-1})$ and $\\mathbb{Q}(\\sqrt{-3})$, then, show that the $m$-th Fourier coefficient of the weighted theta series of those lattices does not vanish, when the shell of norm $m$ of those lattices is not the empty set. Equivalently, we show that the corresponding spherical 2-design does not exist among the nonempty shells in those lattices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-04-09T10:43:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F03",
      "05B30"
    ],
    "keywords": [
      "lehmers conjecture",
      "toy models",
      "weighted theta series",
      "nonempty shells",
      "empty set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.1520B"
    }
  }
}