{
  "id": "2011.12795",
  "version": "v1",
  "published": "2020-11-23T18:39:44.000Z",
  "updated": "2020-11-23T18:39:44.000Z",
  "title": "Super-zeta functions and regularized determinants associated to cofinite Fuchsian groups with finite-dimensional unitary representations",
  "authors": [
    "Joshua S. Friedman",
    "Jay Jorgenson",
    "Lejla Smajlovic"
  ],
  "comment": "submitted to Letters in Mathematical Physics. arXiv admin note: text overlap with arXiv:1607.08053",
  "categories": [
    "math.NT",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $\\chi$ denote a finite dimensional unitary representation of the fundamental group of $M$. Let $\\Delta$ denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over $M$ associated to $\\chi$. From the spectral theory of $\\Delta$, there are three distinct sequences of numbers: The first coming from the eigenvalues of $L^{2}$ eigenfunctions, the second coming from resonances associated to the continuous spectrum, and the third being the set of negative integers. Using these sequences of spectral data, we employ the super-zeta approach to regularization and introduce two super-zeta functions, $\\Z_-(s,z)$ and $\\Z_+(s,z)$ that encode the spectrum of $\\Delta$ in such a way that they can be used to define the regularized determinant of $\\Delta-z(1-z)I$. The resulting formula for the regularized determinant of $\\Delta-z(1-z)I$ in terms of the Selberg zeta function, see Theorem 5.3, encodes the symmetry $z\\leftrightarrow 1-z$, which could not be seen in previous works, due to a different definition of the regularized determinant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-23T18:39:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M36",
      "11F72"
    ],
    "keywords": [
      "regularized determinant",
      "cofinite fuchsian groups",
      "finite-dimensional unitary representations",
      "super-zeta functions",
      "finite dimensional unitary representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}