Relative growth of Dirichlet series with different abscissas of absolute convergence
Abstract
UDC 517.537.72
We study the growth of a Dirichlet series $F(s)=\sum _{n=1}^{\infty}f_n\exp\{s\lambda_n\}$ with zero abscissa of absolute convergence with respect to the entire Dirichlet series $G(s)=\sum _{n=1}^{\infty}g_n\exp\{s\lambda_n\}$ by using the generalized quantities of order $\varrho^0_{\beta,\beta}[F]_G=\varlimsup\nolimits_{\sigma\uparrow 0}\dfrac{\beta(M^{-1}_G(M_F(\sigma)))}{\beta(1/|\sigma|)}$ and lower order $\lambda^0_{\beta,\beta}[F]_G=\varliminf_{\sigma\uparrow 0} \dfrac{\beta(M^{-1}_G(M_F(\sigma)))}{\beta(1/|\sigma|)},$ where $M_F(\sigma)=\sup\{|F(\sigma+it)|\colon t\in{\Bbb R}\},$ $M^{-1}_G(x)$ is the function inverse to $M_G(\sigma),$ and $\beta$ is a positive increasing function growing to $+\infty.$
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