On an estimation of R-type of entire Dirichlet series and its exactness - Hlova, T.Ya.; Filevych, P.V. (Carpathian Mathematical Publications, 2013)
Let $(λ_n)$ be a nonnegative sequence, increasing to $+∞$, $τ=\limsup\limits_{n→∞}\frac{\ln n}{λ_n}$, and $ρ$ be a positive number. It follows from a classical theorem of G. Valiron that for every Dirichlet series of the form $F(s)=\sum a_ne^{sλ_n}$ we have $$\limsup_{σ→+∞}\frac{\ln \sup\{|F(s)|:\,\text{Re}\, s=σ\}}{e^{ρσ}}\le e^{ρτ} \limsup_{n→∞}\frac{λ_n}{eρ}|a_n|^\frac{ρ}{λ_n}.$$ The exactness of this estimation is proved in the paper.