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 .

Let $\Lambda = \{\lambda_{k}\}$ denote a sequence of complex numbers and assume that that the counting function $#\{\lambda_{k} \in \Lambda : | \lambda_{k}| < T\} =O(T^{n})$ for some integer $n$. From Hadamard's theorem, we can construct an entire function $f$ of order at most $n$ such that $\Lambda$ is the divisor $f$. In this article we prove, under reasonably general conditions, that the superzeta function $\Z_{f}(s,z)$ associated to $\Lambda$ admits a meromorphic continuation. Furthermore, we describe the relation between the regularized product of the sequence $z-\Lambda$ and the function $f$ as constructed as a Weierstrass product. In the case $f$ admits a Dirichlet series expansion in some right half-plane, we derive the meromorphic continuation in $s$ of $\Z_{f}(s,z)$ as an integral transform of $f'/f$. We apply these results to obtain superzeta product evaluations of Selberg zeta function associated to finite volume hyperbolic manifolds with cusps.

Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $\phi(s)$ denote the automorphic scattering determinant. From the known functional equation $\phi(s)\phi(1-s)=1$ one concludes that $\phi(1/2)^{2} = 1$. However, except for the relatively few instances when $\phi(s)$ is explicitly computable, one does not know $\phi(1/2)$. In this article we address this problem and prove the following result. Let $N$ and $P$ denote the number of zeros and poles, respectively, of $\phi(s)$ in $(1/2,\infty)$, counted with multiplicities. Let $d(1)$ be the coefficient of the leading term from the Dirichlet series component of $\phi(s)$. Then $\phi(1/2)=(-1)^{N+P} \cdot \mathrm{sgn}(d(1))$.

Let $M$ denote a finite volume, non-compact Riemann surface without elliptic points, and let $B$ denote the Lax-Phillips scattering operator. Using the superzeta function approach due to Voros, we define a Hurwitz-type zeta function $\zeta^{\pm}_{B}(s,z)$ constructed from the resonances associated to $zI -[ (1/2)I \pm B]$. We prove the meromorphic continuation in $s$ of $\zeta^{\pm}_{B}(s,z)$ and, using the special value at $s=0$, define a determinant of the operators $zI -[ (1/2)I \pm B]$. We obtain expressions for Selberg's zeta function and the determinant of the scattering matrix in terms of the operator determinants.