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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Published
13.11.2020
How to Cite
MulyavaO. M., and Sheremeta M. M. “Relative Growth of Dirichlet Series With Different Abscissas of Absolute Convergence”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 11, Nov. 2020, pp. 1535-43, doi:10.37863/umzh.v72i11.6168.
Section
Research articles