  For certain sequences $A$ of positive integers with missing $g$-adic digits,
the Dirichlet series $F_A(s) = \sum_{a\in A} a^{-s}$ has abscissa of
convergence $\sigma_c < 1$. The number $\sigma_c$ is computed. This generalizes
and strengthens a classical theorem of Kempner on the convergence of the sum of
the reciprocals of a sequence of integers with missing decimal digits.
