On the growth of an entire dirichlet series

Abstract

We establish the relation between the increase of the quantityM(σ,F) = ∣a ₀∣ + ∑ ^∞ _n=1 ∣a _n ∣ exp (σλ _n ) and the behavior of sequences (|a _n |) and (λ _n ), where (λ _n ) is a sequence of nonnegative numbers increasing to + ∞, andF(s) =a ₀ + ∑ ^∞ _n=1 a _n e ^sλn,s=σ+it, is the Dirichlet entire series.