{
  "id": "1301.1574",
  "version": "v1",
  "published": "2013-01-08T16:05:26.000Z",
  "updated": "2013-01-08T16:05:26.000Z",
  "title": "On the distribution of eigenvalues of Maass forms on certain moonshine groups",
  "authors": [
    "Jay Jorgenson",
    "Lejla Smajlović",
    "Holger Then"
  ],
  "comment": "A version with higher resolution figures can be downloaded from http://www.maths.bris.ac.uk/~mahlt/research/JST2012a.pdf",
  "categories": [
    "math.NT",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups $\\Gamma_0(N)^+$, where N>1$ is a square-free integer. After we prove that $\\Gamma_0(N)^+$ has one cusp, we compute the constant term of the associated non-holomorphic Eisenstein series. We then derive an \"average\" Weyl's law for the distribution of eigenvalues of Maass forms, from which we prove the \"classical\" Weyl's law as a special case. The groups corresponding to N=5 and N=6 have the same signature; however, our analysis shows that, asymptotically, there are infinitely more cusp forms for $\\Gamma_0(5)^+$ than for $\\Gamma_0(6)^+$. We view this result as being consistent with the Phillips-Sarnak philosophy since we have shown, unconditionally, the existence of two groups which have different Weyl's laws. In addition, we employ Hejhal's algorithm, together with recently developed refinements from [31], and numerically determine the first 3557 of $\\Gamma_0(5)^+$ and the first 12474 eigenvalues of $\\Gamma_0(6)^+$. With this information, we empirically verify some conjectured distributional properties of the eigenvalues.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-08T16:05:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "maass forms",
      "moonshine groups",
      "eigenvalues",
      "conjectured distributional properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.1574J"
    }
  }
}