On the regular growth of Dirichlet series absolutely convergent in a half-plane

Abstract

For the Dirichlet series $F(s) = \sum^{\infty}_{n=1}a_n \exp \{s \lambda_n\}$ with the abscissa of absolute convergence $\sigma a = 0$, conditions on $(λ_n)$ and $(a_n)$ (λn) are established under which $\ln M(\sigma, F) = T_R(1 + o(1)) \exp\{\varrho R/|\sigma|\}$ as $\sigma \uparrow 0$, where$M(σ, F) = \sup\{|F(\sigma + it)| : t \in R\}$ and $T_R$ and $\varrho_R$ are positive constants.