Construction of an Ordinary Dirichlet Series with Convergence beyond the Bohr Strip

TL;DR

This paper constructs explicit examples of Dirichlet series that converge beyond their traditional Bohr strip, advancing understanding of their convergence properties in complex analysis.

Contribution

It provides the first explicit constructions of Dirichlet series that converge beyond their Bohr strip, addressing a longstanding open question.

Findings

Dirichlet series can converge beyond the Bohr strip

Explicit examples demonstrate convergence beyond traditional limits

Advances understanding of convergence regions in complex analysis

Abstract

An ordinary Dirichlet series has three abscissae of interest, describing the maximal regions where the Dirichlet series converges, converges uniformly, and con- verges absolutely. The paper of Hille and Bohnenblust in 1931, regarding the region on which a Dirichlet series can converge uniformly but not absolutely, has prompted much investigation into this region, the "Bohr strip". However, a related natural question has apparently gone unanswered: For a Dirichlet series with non-trivial Bohr strip, how far beyond the Bohr strip might the series converge? We investigate this question by explicit construction, creating Dirichlet series which converge beyond their Bohr strip.