To Sheremeta's theorem on the rate of convergence of positive Dirichlet series (in Ukrainian)

Let $F(z)=\sum_{n=0}^{+\infty} a_n e^{z\lambda_n}$, $a_n\ge 0$ $(n\ge 1)$, be an entire Dirichlet series. We give a~simple proof of the following assertion: if $\int_0^{+\infty} {h(\ln n(t)) }{t^{-2}} dt <+\infty$, then $\sup\left\{\frac{1}{h(\ln n)}\ln\frac{1}{\sigma_n(F)}:\right.$ $\left. n\geq 0\right\}=+\infty,$ where $h(x)$ is a~positive nondecreasing function, $n(t)$ the counting function of the sequence $\lambda_n\uparrow +\infty$, and $\sigma_n(F)=$ \newline $=\max\left\{ \frac 1{\sum_{k=0}^n a_k e^{x\lambda_k}} - \frac 1{F(x)}:x\in \Bbb R\right\}.$