Bohr’s strip for vector valued Dirichlet series

Abstract

Bohr showed that the width of the strip (in the complex plane) on which a given Dirichlet seriesan/n s , s ∈ C, converges uniformly but not absolutely, is at most 1/2, and Bohnenblust-Hille that this bound in general is optimal. We prove that for a given infinite dimensional Banach space Y the width of Bohr's strip for a Dirichlet series with coefficients an in Y is bounded by 1 −1/Cot(Y ), where Cot(Y ) denotes the optimal cotype of Y. This estimate even turns out to be optimal, and hence leads to a new characterization of cotype in terms of vector valued Dirichlet series. Mathematics Subject Classification (2000) Primary 32A05; Secondary 46B07 ·