Vector-valued general Dirichlet series

Abstract

Opened up by early contributions due to, among others, Besicovitch, Bohr, Bohnenblust, Hardy, Hille, Riesz, Neder and Landau, the last 20 years show a substantial revival of systematic research on ordinary Dirichlet series Pann −s , and more recently even on general Dirichlet series Pane −λns . This involves the intertwining of classical work with modern functional analysis, harmonic analysis, in nite dimensional holomorphy and probability theory as well as analytic number theory. Motivated through this line of research the main goal of this article is to start a systematic study of a variety of fundamental aspects of vector-valued general Dirichlet series Pane −λns , so Dirichlet series, where the coe cients are not necessarily in C but in some arbitrary Banach space X.