Super-zeta functions and regularized determinants associated with cofinite Fuchsian groups with finite-dimensional unitary representations
Let M be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $$\chi $$ χ denote a finite dimensional unitary representation of the fundamental group of M . Let $$\Delta $$ Δ denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over M associated with $$\chi $$ χ . From the spectral theory of $$\Delta $$ Δ , there are three distinct sequences of numbers: the first coming from the eigenvalues of $$L^{2}$$ L 2 eigenfunctions, the second coming from resonances associated with 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, $$\mathcal {Z}_-(s,z)$$ Z - ( s , z ) and $$\mathcal {Z}_+(s,z)$$ Z + ( s , z ) that encode the spectrum of $$\Delta $$ Δ in such a way that they can be used to define the regularized determinant of $$\Delta -z(1-z)I$$ Δ - z ( 1 - z ) I . The resulting formula for the regularized determinant of $$\Delta -z(1-z)I$$ Δ - z ( 1 - z ) I in terms of the Selberg zeta function, see Theorem 5.3, encodes the symmetry $$z\leftrightarrow 1-z$$ z ↔ 1 - z .