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.
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The first, second and third authors were supported by MEC and FEDER Project MTM2005-08210.
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Defant, A., García, D., Maestre, M. et al. Bohr’s strip for vector valued Dirichlet series. Math. Ann. 342, 533–555 (2008). https://doi.org/10.1007/s00208-008-0246-z
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DOI: https://doi.org/10.1007/s00208-008-0246-z