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

## TL;DR

This paper develops a framework for meromorphically continuing superzeta functions associated with sequences of complex numbers, linking them to regularized products and applying these results to Selberg zeta functions on hyperbolic manifolds with cusps.

## Contribution

It establishes the meromorphic continuation of superzeta functions for sequences with certain growth conditions and relates these to regularized products, with applications to Selberg zeta functions.

## Key findings

- Superzeta functions admit meromorphic continuation under general conditions.
- Regularized products of sequences are connected to entire functions via Weierstrass products.
- Applications include evaluations of Selberg zeta functions for 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.