Suppose we have a Dirichlet series, and we know that it has an analytic continuation f(s), which has no poles in the region $Re(s) > \sigma$.
Can we say anything about the abscissa of convergence of the series?
(Obvious example: $(1 - 2^{1-s}) \zeta(s) = \sum_n (-1)^n n^{-s}$; the left-hand side is entire, but the right-hand side has abscissa of convergence 0.)