Some Remarks on the Algebra of Bounded Dirichlet Series

TL;DR

This paper investigates the algebraic structure of bounded Dirichlet series, demonstrating the failure of the Corona theorem in this context and refining the Hille-Bohnenblust theorem using probabilistic and deterministic methods.

Contribution

It shows the Corona theorem does not hold for bounded Dirichlet series and provides refined versions of the Hille-Bohnenblust theorem with detailed proofs.

Findings

Corona theorem is false in this setting

Right half-plane is not dense in the maximal ideal space

Refined Hille-Bohnenblust theorem with probabilistic and deterministic proofs

Abstract

We examine the algebra of all Dirichlet Series bounded on the right half plane. We consider the analogue of the Corona theorem in this setting, and show that it is false, i.e. the right half-plane is not dense in the maximal ideal space. We also prove some refinements of the Hille-Bohnenblust theorem, where both probabilistic and deterministic devices are used, and we show how the proof is carried out for each.