The representation of regular functions by Dirichlet series

It is proved that if a system of exponents has the property that any function regular in a closed convex domain $\overline G$ can be represented in an open domain $G$ by a Dirichlet series, then any function regular only in $G$ can be represented in $G$ by a Dirichlet series with the same system of exponents. A study is made of the representation of functions regular in $\overline G$ by Dirichlet series that converge in $\overline G$.