Dirichlet series from the infinite dimensional point of view

Abstract

A classical result of Harald Bohr linked the study of convergent and bounded Dirichlet series on the right half plane with bounded holomorphic functions on the open unit ball of the space $c_0$ of complex null sequences. Our aim here is to show that many questions in Dirichlet series have very natural solutions when, following Bohr's idea, we translate these to the infinite dimensional setting. Some are new proofs and other new results obtained by using that point of view.