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Mathematica Slovaca

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Volume 68, Issue 4


The Riemann hypothesis and universality of the Riemann zeta-function

Ramūnas Garunkštis
  • Institute of Mathematics Faculty of Mathematics and Informatics Vilnius University, Naugarduko 24 LT-03225, Vilnius, Lithuania
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/ Antanas Laurinčikas
  • Institute of Mathematics Faculty of Mathematics and Informatics Vilnius University, Naugarduko 24 LT-03225, Vilnius, Lithuania
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Published Online: 2018-08-06 | DOI: https://doi.org/10.1515/ms-2017-0141


We prove that, under the Riemann hypothesis, a wide class of analytic functions can be approximated by shifts ζ(s + iγk), k ∈ ℕ, of the Riemann zeta-function, where γk are imaginary parts of nontrivial zeros of ζ(s).

MSC 2010: Primary 11M06

Keywords: Riemann hypothesis; Riemann zeta-function; universality; weak convergence


  • [1]

    Bagchi, B.: The Statistical Behaviour and Universality Properties of the Riemann Zeta-Function and Other Allied Dirichlet Series. Ph. D. Thesis, Calcutta, Indian Stat. Institute, 1981.Google Scholar

  • [2]

    Billingsley, P.: Convergence of Probability Measures, Wiley, New York, 1968.Google Scholar

  • [3]

    Bitar, K. M.—Khuri, N. N.—Ren, H. C.: Paths integrals and Voronin’s theorem on the universality of the Riemann zeta-function, Ann. Phys. 211 (1994), 172–196.Google Scholar

  • [4]

    Bui, H. M.—Conrey, B.—Young, M. P.: More than 41% of zeros of zeta function are on the critical line, Acta Arith. 150 (2011), 35–64.CrossrefWeb of ScienceGoogle Scholar

  • [5]

    Conway, J. B.: Functions of One Complex Variable, Springer, Berlin, Heidelberg, New York, 1978.Google Scholar

  • [6]

    Dubickas, A.—Laurinčikas, A.: Distribution modulo 1 and the discrete universality of the Riemann zeta-function, Abh. Math. Semin. Univ. Hambg. 86(1) (2016), 79–87.CrossrefGoogle Scholar

  • [7]

    Garunkštis, R.—Laurinčikas, A.: Discrete mean square of the Riemann zeta-function over imaginary parts of its zeros, Period. Math. Hungar. 76 (2018), 217–228.CrossrefGoogle Scholar

  • [8]

    Gonek, S. M.: Analytic Properties of Zeta and L-Functions, Ph. D. Thesis, University of Michigan, 1979.Google Scholar

  • [9]

    Kuipers, L.—Niederreiter, H.: Uniform Distribution of Sequences. Pure Appl. Math., Wiley-Inter-science, New York, London, Sydney, 1974.Google Scholar

  • [10]

    Laurinčikas, A.: Limit Theorems for the Riemann Zeta-Function, Kluwer, Dordrecht, Boston, London, 1996.Google Scholar

  • [11]

    Mergelyan, S. N.: Uniform approximation to functions of a complex variable, Usp. Matem. Nauk (in Russian) 7(2) (1952), 31–122.Google Scholar

  • [12]

    Reich, A.: Werteverteilung von Zetafunktionen, Arch. Math. 34 (1980), 440–451.CrossrefGoogle Scholar

  • [13]

    Sander, J.—Steuding, J.: Joint universality for sums and products of Dirichlet L-functions, Analysis 26 (2006), 295–312.Google Scholar

  • [14]

    Steuding, J.: Value-Distribution of L-Functions. Lecture Notes in Math. 1877, Springer, Berlin, Heidelberg, New York, 2007.Google Scholar

  • [15]

    Steuding, J.: The roots of the equation ζ(s) = a are uniformly distributed modulo one. In: Anal. Probab. Methods Number Theory, A. Laurinčikas et al. (eds.), TEV, Vilnius, 2012, pp. 243–249.Google Scholar

  • [16]

    Titchmarsh, E. C.: The Theory of the Riemann Zeta-Function, Second edition revised by D. R. Heath-Brown, Clarendon Press, Oxford, 1986.Google Scholar

  • [17]

    Voronin, S. M.: Theorem on the “universality” of the Riemann zeta-function, Izv. Akad. Nauk SSSR (in Russian) 39 (1975), 475–486. Math. USSR Izv. 9 (1975), 443–453.Google Scholar

About the article

Received: 2016-09-08

Accepted: 2017-07-30

Published Online: 2018-08-06

Published in Print: 2018-08-28

Communicated by Federico Pellarin

Citation Information: Mathematica Slovaca, Volume 68, Issue 4, Pages 741–748, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0141.

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