On conditions for Dirichlet series absolutely convergent in a half-plane to belong to the class of convergence

Let M(σ) = sup{|F(σ + it)|: t ∈ ℝ} and μ(σ) = max {|a _n |exp(σλ_n): n ≥ 0}, σ < 0, for a Dirichlet series {fx995-01} with abscissa of absolute convergence σ_a = 0. We prove that the condition ln ln n = o(ln λ_n), n → ∞, is necessary and sufficient for the equivalence of the relations {fx995-02}, for each series of this type. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 851–856, June, 2008.