On an estimation of R-type of entire Dirichlet series and its exactness

Abstract

Let $(\lambda_n)$ be a nonnegative sequence, increasing to $+\infty$, $\tau=\limsup\limits_{n\to\infty}\frac{\ln n}{\lambda_n}$, and $\rho$ be a positive number. It follows from a classical theorem of G. Valiron that for every Dirichlet series of the form $F(s)=\sum a_ne^{s\lambda_n}$ we have

$$\limsup_{\sigma\to+\infty}\frac{\ln \sup\{|F(s)|:\,\text{Re}\, s=\sigma\}}{e^{\rho\sigma}}\le e^{\rho\tau} \limsup_{n\to\infty}\frac{\lambda_n}{e\rho}|a_n|^\frac{\rho}{\lambda_n}.$$

The exactness of this estimation is proved in the paper.