  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 Fr\'echet spaces play a relevant role.
