Analytic Continuation of some zeta functions

TL;DR

This paper explores the methods for extending the domain of various zeta functions through analytic continuation, highlighting applications like deriving asymptotic formulas when natural boundaries are present.

Contribution

It provides an expository overview of meromorphic continuation techniques for zeta functions with and without Euler products, including applications to asymptotic analysis.

Findings

Meromorphic continuation of zeta functions with Euler products

Identification of natural boundaries affecting analytic continuation

Application to asymptotic formulas in number theory

Abstract

This is an expository paper on the meromorphic continuation of zeta functions with Euler products (for example zeta functions of groups and height zeta functions) or without (for example the Goldbach zeta function). As an application we show how a natural boundary of analytic continuation can give asymptotic results.