Abstract
Voronin’s theorem states that the Riemann zeta-function ζ(s) is universal in the sense that all analytic functions that are defined and have no zeros on the right half of the critical strip can be approximated by the shifts ζ(s + iτ), τ ∈ ℝ. Some results on the approximation by the shifts ζ(s + iϕ(τ)) with some function ϕ(τ) are also known. In this paper, it is established that an analytic function without zeros in the strip 1/2 + 1/(2α) < Res < 1 can be approximated by the shifts ζ(s + i logατ) with α > 1.
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References
H. Bohr and R. Courant, “Neue Anwendungen der Theorie der Diophantischen Approximationen auf Riemannschen Zetafunktion,” J. Reine Angew. Math. 144, 249–274 (1914).
S. M. Voronin, “The distribution of the nonzero values of the Riemann ζ-function,” in Trudy Mat. Inst. Steklova (MIAN, Moscow, 1972), Vol. 128, pp. 131–150 [Proc. Steklov Inst. Math. 128, 153–175 (1972)].
S. M. Voronin, “Theorem on the “universality” of the Riemann zeta-function,” Izv. Akad. Nauk SSSR Ser. Mat. 39 (3), 475–486 (1975) [Math. USSR-Izv. 9 (3), 443–453 (1975)].
B. Bagchi, Statistical Behavior and Universality Properties of the Riemann Zeta-Function and Other Allied Dirichlet Series, Thesis (Indian Statistical Institute, Calcutta, 1981).
A. Laurinčikas, Limit Theorems for the Riemann Zeta-Function, in Math. Appl. (Kluwer Acad. Publ., Dordrecht, 1996), Vol. 352.
S.M. Voronin and A. A. Karatsuba, The Riemann Zeta-Function (Fizmatlit, Moscow, 1994) [in Russian].
A. Laurinčikas and L. Meshka, “Sharpening of the universality inequality,” Mat. Zametki 96 (6), 905–910 (2014) [Math. Notes 96 (6), 971–976 (2014)].
J.-L. Mauclaire, “Universality of the Riemann zeta-function: two remarks,” Ann. Univ. Sci. Budapest. Sec. Comput. 39, 311–319 (2013).
Ł. Pańkowski, “Joint universality for dependent L-functions,” Ramanujan J. 45 (1), 181–195 (2018).
A. Laurinčikas, R. Macaitienć, and D. Šiaučiūnas, “A generalization of Voronin’s theorem,” Lith. Math. J. 59 (2), 156–168 (2019).
A. Ivič, The Riemann Zeta-Function. The Theory of the Riemann Zeta-Functionwith Applications (John Wiley & Sons, New York, 1985).
P. Billingsley, Convergence of Probability Measures (J. Wiley, New York–London, 1968; Nauka, Moscow, 1977).
S. M. Mergelyan, “Uniform approximations of functions of a complex variable,” Uspekhi Mat. Nauk 7 (2 (48)), 31–122 (1952) [Am. Math. Soc. Transl. 101 (1954)].
Acknowledgments
The author wishes to express gratitude to the referee for useful remarks.
Funding
The research is funded by the European Social Fund according to the activity “Improvement of Researchers’ qualification by implementing world-class R&D projects” of Measure No. 09.3.3-LMTK712-01-0037.
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Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 3, pp. 400–411.
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Laurinčikas, A. On a Generalization of Voronin’s Theorem. Math Notes 107, 442–451 (2020). https://doi.org/10.1134/S0001434620030086
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DOI: https://doi.org/10.1134/S0001434620030086