Weighted discrete universality of the Riemann zeta-function
Abstract
It is well known that the Riemann zeta-function is universal in the Voronin sense, i.e., its shifts ζ(s + iτ), τ ∈ R, approximate a wide class of analytic functions. The universality of ζ(s) is called discrete if τ take values from a certain discrete set. In the paper, we obtain a weighted discrete universality theorem for ζ(s) when τ takes values from the arithmetic progression {kh : k ∈N} with arbitrary fixed h > 0. For this, two types of h are considered.
Keyword : approximation of analytic functions, Mergelyan theorem, Riemann zeta-function, universality, weak convergence
How to Cite
Laurinčikas, A., Šiaučiūnas, D., & Vadeikis, G. (2020). Weighted discrete universality of the Riemann zeta-function. Mathematical Modelling and Analysis, 25(1), 21-36. https://doi.org/10.3846/mma.2020.10436
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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R. Macaitienė, M. Stoncelis and D. Šiaučiūnas. A weighted universality theorem for periodic zeta-functions. Math. Modell. Analysis, 22(1):95–105, 2017.
https://doi.org/10.3846/13926292.2017.1269373
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A. Reich. Werteverteilung von zetafunktionen. Arch. Math., 34(1):440–451, 1980. https://doi.org/10.1007/BF01224983
A.N. Shiryaev. Probability. Graduate Texts Math. vol. 95, Springer-Verlag, New York, 1984.
J. Steuding. Value-Distribution of L-Functions. Lecture Notes Math. vol. 1877, Springer, Berlin, Heidelberg, 2007.
S.M. Voronin. On the distribution of nonzero values of the Riemann ζ-function. Trudy Mat. Inst. Steklov, 128:131–150, 1972 (in Russian).
S.M. Voronin. Theorem on the “universality” of the Riemann zeta-function. Izv. Akad. Nauk SSSR, Ser. Matem., 39:475–486, 1975 (in Russian).
P. Billingsley. Convergence of Probability Measures. Willey, New York, 1968.
K.M. Bitar, N.N. Khuri and H.C. Ren. Path integrals and Voronin’s theorem on the universality of the Riemann zeta-function. Ann. Phys., 211(1):172–196, 1991. https://doi.org/10.1016/0003-4916(91)90196-F
H. Bohr. Über das verhalten von ζ(s) in der halbebene σ > 1. Nachr. Akad. Wiss. G¨ottingen II Math. Phys. Kl., pp. 409–428, 1911.
H. Bohr and R. Courant. Neue anwendungen der theorie der diophantischen approximationen auf die Riemannsche zetafunktion. Reine Angew. Math., 1914(144):249–274, 1914. https://doi.org/10.1515/crll.1914.144.249
A. Dubickas and A. Laurinčikas. Distribution modulo 1 and the discrete universality of the Riemann zeta-function. Abh. Math. Semin. Univ. Hamb., 86(1):79– 87, 2016. https://doi.org/10.1007/s12188-016-0123-8
R. Garunkštis and A. Laurinčikas. Riemann hypothesis and universality of the Riemann zeta-function. Math. Slovaca, 68(4):741–748, 2018. https://doi.org/10.1515/ms-2017-0141
R. Garunkštis, A. Laurinčikas and R. Macaitienė. Zeros of the Riemann zeta-function and its universality. Acta Arith., 181:127–142, 2017. https://doi.org/10.4064/aa8583-5-2017
S.M. Gonek. Analytic Properties of Zeta and L-Functions. PhD Thesis, University of Michigan, 1979.
A. Laurinčikas. On the universality of the Riemann zeta-function. Lith. Math. J., 35(4):399–402, 1995. https://doi.org/10.1007/BF02335599
A. Laurinčikas. Limit Theorems for the Riemann Zeta-Function. Kluwer, Dordrecht, 1996.
A. Laurinčikas. On the Matsumoto zeta-function. Acta Arith., 84:1–16, 1998. https://doi.org/10.4064/aa-84-1-1-16
A. Laurinčikas. Discrete universality of the Riemann zeta-function and uniform distribution modulo 1. St. Petersburg Math. J., 30:103–110, 2019. https://doi.org/10.1090/spmj/1532
A. Laurinčikas, K. Matsumoto and J. Steuding. Discrete universality of L-functions of new forms. II. Lith. Math. J., 56(2):207–218, 2016. https://doi.org/10.1007/s10986-016-9314-3
A. Laurinčikas and J. Petuškinaitė. Universality of Dirichlet L-functions and non-trivial zeros of the Riemann zeta-function. Sb. Math., 210, 2019. https://doi.org/10.1070/SM9194
R. Macaitienė. On discrete universality of the Riemann zeta-function with respect to uniformly distributed shifts. Arch. Math., 108(3):271–281, 2017. https://doi.org/10.1007/s00013-016-0998-8
R. Macaitienė, M. Stoncelis and D. Šiaučiūnas. A weighted discrete universality theorem for periodic zeta-functions. II. Math. Modell. Analysis, 22(6):750–762, 2017. https://doi.org/10.3846/13926292.2017.1365779
R. Macaitienė, M. Stoncelis and D. Šiaučiūnas. A weighted universality theorem for periodic zeta-functions. Math. Modell. Analysis, 22(1):95–105, 2017.
https://doi.org/10.3846/13926292.2017.1269373
S.N. Mergelyan. Uniform approximations to functions of complex variable. Usp. Mat. Nauk., 7:31–122, 1952 (in Russian).
H.L. Montgomery. Topics in Multiplicative Number Theory. Lecture Notes Math. Vol. 227, Springer-Verlag, Berlin, 1971. https://doi.org/10.1007/BFb0060851
L . Pańkowski. Joint universality for dependent L-functions. Ramanujan J., 45(1):181–195, 2018. https://doi.org/10.1007/s11139-017-9886-5
A. Reich. Werteverteilung von zetafunktionen. Arch. Math., 34(1):440–451, 1980. https://doi.org/10.1007/BF01224983
A.N. Shiryaev. Probability. Graduate Texts Math. vol. 95, Springer-Verlag, New York, 1984.
J. Steuding. Value-Distribution of L-Functions. Lecture Notes Math. vol. 1877, Springer, Berlin, Heidelberg, 2007.
S.M. Voronin. On the distribution of nonzero values of the Riemann ζ-function. Trudy Mat. Inst. Steklov, 128:131–150, 1972 (in Russian).
S.M. Voronin. Theorem on the “universality” of the Riemann zeta-function. Izv. Akad. Nauk SSSR, Ser. Matem., 39:475–486, 1975 (in Russian).