GENERALIZED AND MODIFIED ORDERS OF GROWTH FOR DIRICHLET SERIES ABSOLUTELY CONVERGENT IN A HALF-PLANE.

Let λ = (λn)n∈N0 be a non-negative sequence increasing to +∞, τ (λ) = limn→∞(ln n/λn), and D0(λ) be the class of all Dirichlet series of the form F(s) = P∞ n=0 an(F)esλn absolutely convergent in the half-plane Re s < 0 with an(F) ̸= 0 for at least one integer n ≥ 0. Also, let α be a continuous function on [x0,+∞) increasing to +∞, β be a continuous function on [a, 0) such that β(σ) → +∞ as σ ↑ 0, and γ be a continuous positive function on [b, 0). In the article, we investigate the growth of a Dirichlet series F ∈ D0(λ) depending on the behavior of the sequence (|an(F)|) in terms of its α, β, γ-orders determined by the equalities... where μ(σ) = max{|an(F)|eσλn: n ≥ 0} and M(σ) = sup{|F(s)|: Re s = σ} are the maximal term and the supremum modulus of the series F, respectively. In particular, if for every fixed t > 0 we have α(tx) ∼ α(x) as x → +∞, β(tσ) ∼ t-ρβ(σ) as σ ↑ 0 for some fixed ρ > 0, 0 < limσ↑0 γ(tσ)/γ(σ) ≤ limσ↑0 γ(tσ)/γ(σ) < +∞, Φ(σ) = α-1(β(σ))/γ(σ) for all σ ∈ [σ0, 0), eΦ(x) = max{xσ -Φ(σ): σ ∈ [σ0, 0)} for all x ∈ R, and ΔΦ(λ) = limn→∞(-ln n/eΦ(λn)), then: (a) for each Dirichlet series F ∈ D0(λ) we have... (b) if τ (λ) > 0, then for each p0 ∈ [0,+∞] and any positive function Ψ on [c, 0) there exists a Dirichlet series F ∈ D0(λ) such that R∗ α,β,γ(F) = p0 and M(σ, F) ≥ Ψ(σ) for all σ ∈ [σ0, 0); (c) if τ (λ) = 0, then (Rα,β,γ(F))1/ρ ≤ (R∗ α,β,γ(F))1/ρ + ΔΦ(λ) for every Dirichlet series F ∈ D0(λ); (d) if τ (λ) = 0, then for each p0 ∈ [0,+∞] there exists a Dirichlet series F ∈ D0(λ) such that R∗ α,β,γ(F) = p0 and (Rα,β,γ(F))1/ρ = (R∗ α,β,γ(F))1/ρ + ΔΦ(λ). [ABSTRACT FROM AUTHOR]