A Comparison Test for Meromorphic Extensions

TL;DR

This paper introduces a comparison test for meromorphic extensions of series, showing that closeness implies similar extension properties, and provides new examples and counterexamples related to Dirichlet series.

Contribution

It presents a novel comparison test for meromorphic extensions and demonstrates its optimality with counterexamples, advancing understanding of Dirichlet series extensions.

Findings

The comparison test guarantees similar meromorphic extensions for close series.

New examples of Dirichlet series with meromorphic extensions are constructed.

Counterexamples show the limits of the comparison test, with series close but not extendable.

Abstract

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.