{
  "id": "0812.4643",
  "version": "v2",
  "published": "2008-12-26T05:47:25.000Z",
  "updated": "2009-04-14T08:28:30.000Z",
  "title": "Toy models for D. H. Lehmer's conjecture",
  "authors": [
    "Eiichi Bannai",
    "Tsuyoshi Miezaki"
  ],
  "comment": "12 pages",
  "journal": "J. Math. Soc. Japan 62 (2010), no. 3, 687-705",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "In 1947, Lehmer conjectured that the Ramanujan $\\tau$-function $\\tau (m)$ never vanishes for all positive integers $m$, where the $\\tau (m)$ are the Fourier coefficients of the cusp form $\\Delta_{24}$ of weight 12. Lehmer verified the conjecture in 1947 for $m<214928639999$. In 1973, Serre verified up to $m<10^{15}$, and in 1999, Jordan and Kelly for $m<22689242781695999$. The theory of spherical $t$-design, and in particular those which are the shells of Euclidean lattices, is closely related to the theory of modular forms, as first shown by Venkov in 1984. In particular, Ramanujan's $\\tau$-function gives the coefficients of a weighted theta series of the $E_{8}$-lattice. It is shown, by Venkov, de la Harpe, and Pache, that $\\tau (m)=0$ is equivalent to the fact that the shell of norm $2m$ of the $E_{8}$-lattice is an 8-design. So, Lehmer's conjecture is reformulated in terms of spherical $t$-design. Lehmer's conjecture is difficult to prove, and still remains open. In this paper, we consider toy models of Lehmer's conjecture. Namely, we show 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 5 (resp. 7)-design does not exist among the shells in the $\\mathbb{Z}^2$-lattice (resp. $A_{2}$-lattice).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-04-14T08:28:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F03",
      "05B30"
    ],
    "keywords": [
      "lehmers conjecture",
      "toy models",
      "weighted theta series",
      "th fourier coefficient",
      "empty set"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0812.4643B"
    }
  }
}