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

Abstract

An ordinary Dirichlet series has three abscissae of interest, describing the maximal regions where the Dirichlet series converges, converges uniformly, and converges 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.