, Volume 342, Issue 3, pp 533-555
Date: 30 May 2008

Bohr’s strip for vector valued Dirichlet series

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Abstract

Bohr showed that the width of the strip (in the complex plane) on which a given Dirichlet series \({\sum a_n/ n^s, \, s \in \mathbb{C}}\) , 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 a n 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.

The first, second and third authors were supported by MEC and FEDER Project MTM2005-08210.