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On conditions for Dirichlet series absolutely convergent in a half-plane to belong to the class of convergence

Ukrainian Mathematical Journal Aims and scope

Let M(σ) = sup{|F(σ + it)|: t ∈ ℝ} and µ(σ) = max {|a n |exp(σλn): n ≥ 0}, σ < 0, for a Dirichlet series {fx995-01} with abscissa of absolute convergence σa = 0. We prove that the condition ln ln n = o(ln λn), n → ∞, is necessary and sufficient for the equivalence of the relations {fx995-02}, for each series of this type.

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References

  1. V. S. Boichuk, On the Growth of Dirichlet Series Absolutely Convergent in a Half-Plane [in Russian], Naukova Dumka, Kiev (1976).

    Google Scholar 

  2. Yu. M. Gal' and M. N. Sheremeta, “On analytic functions belonging to the class of convergence,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 7, 11–14 (1985).

  3. O. M. Mulyava, “On convergence classes of Dirichlet series,” Ukr. Mat. Zh., 51, No. 11, 1485–1494 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  4. O. M. Sumyk and M. M. Sheremeta, “On connection between the growth of maximum modulus and maximal term of entire Dirichlet series in terms of m-member asymptotics,” Mat. Stud., 19, No. 1, 83–88 (2003).

    MATH  MathSciNet  Google Scholar 

  5. A. F. Leont'ev, Exponential Series [in Russian], Nauka, Moscow (1976).

    MATH  Google Scholar 

  6. M. M. Sheremeta, Entire Dirichlet Series [in Ukrainian], ISDO, Kyiv (1993).

    Google Scholar 

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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 851–856, June, 2008.

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Mulyava, O.M., Sheremeta, M.M. On conditions for Dirichlet series absolutely convergent in a half-plane to belong to the class of convergence. Ukr Math J 60, 995–1002 (2008). https://doi.org/10.1007/s11253-008-0097-5

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  • DOI: https://doi.org/10.1007/s11253-008-0097-5

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