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Relative Growth of Dirichlet Series with Different Abscissas of Absolute Convergence

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Ukrainian Mathematical Journal Aims and scope

We study the growth of a Dirichlet series \( F(s)={\sum}_{n=1}^{\infty }{f}_n\exp \left\{s{\uplambda}_n\right\} \) with zero abscissa of absolute convergence with respect to the entire Dirichlet series \( G(s)={\sum}_{n=1}^{\infty }{g}_n\exp \left\{s{\uplambda}_n\right\} \) by using generalized quantities of an order \( {\upvarrho}_{\beta, \beta}^0{\left[F\right]}_G=\lim\ {\sup}_{\sigma \uparrow 0}\frac{\beta \left({M}_G^{-1}\left({M}_F\left(\sigma \right)\right)\right)}{\beta \left(1/\left|\sigma \right|\right)} \) and a lower order \( {\uplambda}_{\beta, \beta}^0{\left[F\right]}_G=\lim\ {\operatorname{inf}}_{\sigma \uparrow 0}\frac{\beta \left({M}_G^{-1}\left({M}_F\left(\sigma \right)\right)\right)}{\beta \left(1/\left|\sigma \right|\right)}, \) where \( {M}_F\left(\sigma \right)=\sup \left\{\left|F\left(\sigma + it\right)\right|:t\in \mathrm{\mathbb{R}}\right\},{M}_G^{-1}(x) \) is the function inverse to MG(σ), and β is a positive function increasing to +∞.

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Authors and Affiliations

  1. Kyiv National University of Food Technologies, Kyiv, Ukraine

    O. M. Mulyava

  2. I. Franko Lviv National University, Lviv, Ukraine

    M. M. Sheremeta

Authors
  1. O. M. Mulyava
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  2. M. M. Sheremeta
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Correspondence to M. M. Sheremeta.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 11, pp. 1535–1543, November, 2020. Ukrainian DOI: 10.37863/umzh.v72i11.6168.

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Mulyava, O.M., Sheremeta, M.M. Relative Growth of Dirichlet Series with Different Abscissas of Absolute Convergence. Ukr Math J 72, 1771–1783 (2021). https://doi.org/10.1007/s11253-021-01887-1

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  • DOI: https://doi.org/10.1007/s11253-021-01887-1

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