Abscissas of weak convergence of vector valued Dirichlet series

The abscissas of convergence, uniform convergence and absolute convergence of vector valued Dirichlet series with respect to the original topology and with respect to the weak topology sigma(X, X') of a locally convex space X, in particular of a Banach space X, are compared. The relation of their coincidence with geometric or topological properties of the underlying space X is investigated. Cotype in the context of Banach spaces, and nuclearity and certain topological invariants for Frechet spaces play a relevant role. (C) 2015 Elsevier Inc. All rights reserved.