Title: Superzeta functions, regularized products, and the Selberg zeta function on hyperbolic manifolds with cusps,
Abstract: 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.