On the relative growth of Dirichlet series with zero abscissa of absolute convergence - O. M. Mulyava (Математичні Студії, 2021)
Let $F$ and $G$ be analytic functions given by Dirichlet series with exponents increasing to $+∞$ and zero abscissa of absolute convergence. The growth of $F$ with respect to $G$ is studied through the generalized order $$ρ^0_{α,β}[F]_G=\varlimsup\limits_{σ\uparrow 0}\dfrac{α(1/|M^{-1}_G(M_F(σ)|)}{β(1/|σ|)}$$ and the generalized lower order $$λ^0_{α,β}[F]_G=\varliminf\limits_{σ\uparrow 0} \dfrac{α(1/|M^{-1}_G(M_F(σ)|)}{β(1/|σ|)},$$ where $M_F(σ)=\sup\{|F(σ+it)|:\,t\in{\mathbb R}\},$ $M^{-1}_G(x)$ is the function inverse to $M_G(σ)$ and $α$ and $β$ are positive increasing to $+∞$ functions. Formulas are found for the finding these quantities.