A Comparison Test for Meromorphic Extensions

We provide a comparison test for meromorphic extensions, i.e., if two series are close enough" then the existence of a meromorphic extension of one to the entire complex plane ensures a similar extension for the other. We use this result to generate new examples of Dirichlet series admitting meromorphic extensions. Moreover, we demonstrate that our requirements are optimal by constructing a collection of counterexamples where the series are close but not enough": one series admits a meromorphic extension while the other possesses a natural boundary.