Remarks to the growth of the maximum modulus of Dirichlet series

Abstract

For a Dirichlet series $F(s)=\sum\limits_{n=0}^{\infty} a_n\exp\{s\lambda_n\},\, s=\sigma+it$, with the abscissa of absolute convergence $\sigma_a=A\in(-\infty,\,+\infty]$, let $ M(\sigma, F)=\sup\{|F(\sigma+it)|:\,t\in {\mathbb R}\}$ for $\sigma<A$. By $L$ we denote a class of continuous non-negative on $(-\infty,\,+\infty)$ functions $\alpha$ such that $\alpha(x)=\alpha(x_0)\ge 0$ for $x\le x_0$ and $\alpha(x)\uparrow+\infty$ as $x_0\le x\to+\infty$. It is proved, for example, that if $A=+\infty$, $p>1$, $q>1$, $\alpha \in L$, $\beta \in L$, $\ln\,\beta(x+O(1))=(1+o(1))\ln\,\beta(x)$ as $x\to+\infty$ and $\varlimsup\limits_{n\to\infty}\frac{\ln\,\ln\,\alpha(\lambda_n)}{\ln\,\beta\left(\frac{1}{\lambda_n}\ln\,\frac{1}{|a_n|}\right)}=\eta^*>0$, then $\varlimsup\limits_{\sigma\to+\infty}\frac{\alpha(\ln\,M(\beta^{-1}(q\beta(\sigma)),F))}{\alpha^p(\ln\,M(\sigma,F))}=+\infty$ for each $q>p^{1/\eta^*}$. Similar result is obtained for Dirichlet series with zero abscissa absolute convergence.