On regular variation of entire Dirichlet series

Consider an entire (absolutely convergent in $\mathbb{C}$) Dirichlet series $F$ with the exponents $\lambda_n$, i.e., of form $F(s)=\sum_{n=0}^\infty a_ne^{s\lambda_n}$, and, for all $\sigma\in\mathbb{R}$, put $\mu(\sigma,F)=\max\{|a_n|e^{\sigma\lambda_n}:n\ge0\}$ and $M(\sigma,F)=\sup\{|F(s)|:\operatorname{Re}s=\sigma\}$. Previously, first authors M.M.~Sheremeta proved that if $\omega(\lambda)<C(\rho)$, then regular variation function $\ln\mu(\sigma,F)$ index $\rho$ implies $\ln M(\sigma,F)$ $\rho$, constructed examples $F$, which is a regularly varying not $\rho$. For we have $\lambda_n=\ln\ln n$ $n\ge n_0$ case $\rho=1$, $\lambda_n\sim(\ln n)^{(\rho-1)/\rho}$ as $n\to\infty$ $\rho>1$. In present article prove same property can arbitrary sequence $\lambda=(\lambda_n)_{n=0}^\infty$ satisfying $\omega(\lambda)<C(\rho)$. More precisely, $\omega(\lambda)\ge C(\rho)$, there exists $\Phi(\sigma)$ such that, positive $l(\sigma)$ on $[a,+\infty)$, \mu(\sigma, F)\sim\Phi(\sigma)$ $\sigma\to+\infty$ $M(\sigma,F)\ge l(\sigma)$ $\sigma\ge\sigma_0$.