The growth of the maximal term of Dirichlet series

Abstract

Let $\Lambda$ be the class of nonnegative sequences $(\lambda_n)$ increasing to $+\infty$, $A\in(-\infty,+\infty]$, $L_A$ be the class of continuous functions increasing to $+\infty$ on $[A_0,A)$, $(\lambda_n)\in\Lambda$, and $F(s)=\sum a_ne^{s\lambda_n}$ be a Dirichlet series such that its maximum term $\mu(\sigma,F)=\max_n|a_n|e^{\sigma\lambda_n}$ is defined for every $\sigma\in(-\infty,A)$. It is proved that for all functions $\alpha\in L_{+\infty}$ and $\beta\in L_A$ the equality $$\rho^*_{\alpha,\beta}(F)=\max_{(\eta_n)\in\Lambda}\overline{\lim_{n\to\infty}}\frac{\alpha(\eta_n)}{\beta\left(\frac{\eta_n}{\lambda_n}+\frac{1}{\lambda_n}\ln\frac{1}{|a_n|}\right)}$$ holds, where $\rho^*_{\alpha,\beta}(F)$ is the generalized $\alpha,\beta$-order of the function $\ln\mu(\sigma,F)$, i.e. $\rho^*_{\alpha,\beta}(F)=0$ if the function $\mu(\sigma,F)$ is bounded on $(-\infty,A)$, and $\rho^*_{\alpha,\beta}(F)=\overline{\lim_{\sigma\uparrow A}}\alpha(\ln\mu(\sigma,F))/\beta(\sigma)$ if the function $\mu(\sigma,F)$ is unbounded on $(-\infty,A)$.