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

Superzeta functions, regularized products, and the Selberg zeta function on hyperbolic manifolds with cusps