On the domain of absolute convergence of Dirichlet series in several variables

Abstract

with complex Am and Umk. In ?1 we show that the domain of absolute convergence of this series is convex. Conversely, given a convex domain in E2n=E2n(xl, Yl, , * Xn, yn) and a preassigned sequence of exponents {mk } we construct a Dirichlet series of the given type with the given domain as its domain of absolute convergence, provided the sequence { 0mk } satisfies a certain set of conditions. In ?2 we construct a maximal domain of ordinary convergence. For one complex variable this reduces to a simplified construction of the maximal domain of convergence as given by Hille. We also find a relationship between the domain of absolute convergence and the maximal domain of convergence. For one complex variable this relationship reduces to a refinement of a result by Gallie [2] and for one variable with positive increasing exponents it reduces to the classical theorem regarding the relationship between the abscissas of absolute and ordinary convergence.