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A Comparison Test for Meromorphic Extensions

Adi Glücksam, Yuzhou Joey Zou

TL;DR

This work develops a comparison type criterion for meromorphic extensions of Dirichlet series: if a base series $D(s)=\sum a_n c_n^s$ with $c_n\to0$ extends meromorphically to $\mathbb C$ and a perturbation $\tilde c_n$ arises from a function $g$ with a controlled asymptotic expansion, then the perturbed series $\sum a_n \tilde c_n^s$ also extends meromorphically to $\mathbb C$. The authors prove a positive result under a precise asymptotic expansion for the perturbation $h$ in $\tilde c_n=c_n^{\sigma}(1+h(c_n))$, decomposing the expansion into a main term and a holomorphic error to construct a sequence of meromorphic approximants $g_N$ that converge to the global extension; they also provide a sharp counterexample showing that the expansion condition is essentially necessary. The paper includes numerous examples such as almost geometric progressions, almost zeta functions, and rough perturbations to illustrate applicability, and proves that polynomially bounded perturbations can create natural boundaries, thus delineating the method’s limits. Overall, the results offer a robust framework for generating new Dirichlet series with meromorphic extensions by controlled perturbations, clarifying the interplay between base perturbations and analytic continuation.

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. We demonstrate that our requirements are optimal: we construct 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.

A Comparison Test for Meromorphic Extensions

TL;DR

This work develops a comparison type criterion for meromorphic extensions of Dirichlet series: if a base series with extends meromorphically to and a perturbation arises from a function with a controlled asymptotic expansion, then the perturbed series also extends meromorphically to . The authors prove a positive result under a precise asymptotic expansion for the perturbation in , decomposing the expansion into a main term and a holomorphic error to construct a sequence of meromorphic approximants that converge to the global extension; they also provide a sharp counterexample showing that the expansion condition is essentially necessary. The paper includes numerous examples such as almost geometric progressions, almost zeta functions, and rough perturbations to illustrate applicability, and proves that polynomially bounded perturbations can create natural boundaries, thus delineating the method’s limits. Overall, the results offer a robust framework for generating new Dirichlet series with meromorphic extensions by controlled perturbations, clarifying the interplay between base perturbations and analytic continuation.

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. We demonstrate that our requirements are optimal: we construct 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.
Paper Structure (14 sections, 10 theorems, 67 equations)

This paper contains 14 sections, 10 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Some History
  3. Results
  4. Structure of the Paper
  5. Acknowledgements
  6. Preliminaries
  7. The proof of Theorem \ref{['thm:extension']}
  8. Examples and Counterexamples
  9. On the Usefulness of Theorem \ref{['thm:extension']}- Examples:
  10. On Almost Geometric Progressions:
  11. An Almost Zeta Function
  12. Another interesting example
  13. A rough perturbation- still good enough:
  14. On the Sharpness of Theorem \ref{['thm:extension']}- The Proof of Theorem \ref{['thm:example']}:

Key Result

Theorem 1.2

Let $D(s):=\underset{{n} = {1}}{\overset{\infty}\sum}\; a_n\cdot c_n^s$ be a Dirichlet series with positive bases $c_n$ that converges absolutely in the half-plane $\mathcal{P}_0=\left\{{\mathop{\mathrm{Re}}\nolimits(s)>S_0}\right\}$, and extends meromorphically to $\mathbb C$. Suppose $h:B(0,\varep where $\{\alpha_j\}\subset\mathbb C$ and $\left\{{\sigma_j}\right\}\subset(0,\infty)$ satisfies $\s

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • proof
  • Lemma 2.3
  • proof
  • ...and 16 more