Estimates of generalized orders of Dirichlet series (Q2768813)

Let \(0=l_0<l_{n}\uparrow+\infty,\;n\to\infty\) and the Dirichlet series \(F(s)=\sum\limits_{n=0}^{\infty}a_{n}\exp\{sl_{n}\}\), \(s=\sigma+it\) has abscissa of absolute convergence \(A\in(-\infty;+\infty]\). For \(\sigma<A\) let us denote \(M(\sigma,F)=\sup\{|F(\sigma+it)|: t\in \mathbb{R}\}\) and let \(L\) be a class of continuous slowly increasing functions. If \(F(s)\) is an entire function, \(\alpha\in L,\;\beta\in L\), then \(\rho_{\alpha\beta}(F)=\mathop{\overline{\lim}}\limits_{\sigma\to+\infty}\frac{\alpha(\ln M(\sigma,F))}{\beta(\sigma)}\) is called generalized order of the series \(F(s)\). If \(A=0\), then \(\rho^0_{\alpha\beta}(F)=\mathop{\overline{\lim}}\limits_{\sigma\to 0}\frac{\alpha(\ln M(\sigma,F))}{\beta(1/\sigma)}\) is called generalized order of the series \(F(s)\). The author obtains estimates and relations for generalized orders depending on the values of \(A\).