View More View Less
  • 1 Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko str. 24, LT-03225 Vilnius, Lithuania
Restricted access

Purchase article

USD  $25.00

1 year subscription (Individual Only)

USD  $800.00

Abstract

Let 0 < γ1 < γ2 < ··· ⩽ ··· be the imaginary parts of non-trivial zeros of the Riemann zeta-function. In the paper, we consider the approximation of analytic functions by shifts of the Hurwitz zeta-function ζ(s + kh, α), h > 0, with parameter α such that the set {log(m + α): m0} is linearly independent over the field of rational numbers. For this, a weak form of the Montgomery conjecture on the pair correlation of {γk} is applied.

  • [1]

    Bagchi, B., The statistical behaviour and universality properties of the Riemann zeta function and other allied Dirichlet series, Ph. D. Thesis, Indian Statistical Institute, Calcutta (1981).

    • Search Google Scholar
    • Export Citation
  • [2]

    Billingsley, P., Convergence of Probability Measures, Wiley, New York (1968).

  • [3]

    Cassels, J. W. S., Footnote to a note of Davenport and Heilbronn, J. London Math. Soc, 36 (1961), 177184.

  • [4]

    Davenport, H. and Heilbronn, H., On the zeros of certain Dirichlet series, J. London Math. Soc, 11 (1936), 181185.

  • [5]

    Gonek, S. M., Analytic properties of zeta and L-functions, Ph. D. Thesis, University of Michigan (1979).

  • [6]

    Hurwitz, A., Einige Eigenschaften der Dirichlet’schen FunktionenF ( s ) = ( D n ) 1 n s, die bei der Bestimmung der Klassenanzahlen binärer quadratischer Formen auftreten, Z. Math. Physik, 27 (1882), 86101.

    • Search Google Scholar
    • Export Citation
  • [7]

    Kuipers, L. and Niederreiter, H., Uniform Distribution of Sequences, Pure and Applied Math., Wiley-Interscience, New York, London, Sydney (1974).

    • Search Google Scholar
    • Export Citation
  • [8]

    Laurinčikas, A., Limit Theorems for the Riemann Zeta-Function, Kluwer Academic Publishers, Dordrecht, Boston, London (1996).

  • [9]

    Laurinčikas, A., The joint universality of Hurwitz zeta-functions, Siauliai Math.ˇ Semin., 3(11 ) (2008), 169187.

  • [10]

    Laurinčikas, A., A discrete universality theorem for the Hurwitz zeta-function, J. Number Theory, 143 (2014), 232247.

  • [11]

    Laurinčikas, A., On discrete universality of the Hurwitz zeta-function, Results Math. 72 (2017) no.1–2, 907–917.

  • [12]

    Laurinčikas, A. and Garunkˇstis, R., The Lerch Zeta-Function, Kluwer Academic Publishers, Dordrecht, Boston, London (2002).

  • [13]

    Laurinčikas, A.ˇ and Macaitiene, R.˙ , The discrete universality of the periodic Hurwitz zeta-function, Integral Tranforms Spec. Funct., 20 (2009) no. 9–10, 673686.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [14]

    Laurinčikas, A.ˇ and Meśka, L., On the modification of the universality of Hurwitz zeta-functions, Nonlinear Analysis: Model. Control, 21 (2016) no. 4, 564576.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [15]

    Matsumoto, K., A survey on the theory of universality for zeta and L-functions, in: Number Theory: Plowing and Starring Through High Wave Forms (eds. M. Kaneko ), Proc. 7th China-Japan Seminar, Fukuoka, 2013, Series Number Theory and Appl., vol. 11, World Sci. Publ. Co. (2015), 95144.

    • Search Google Scholar
    • Export Citation
  • [16]

    Mergelyan, S. N., Uniform approximations of functions of a complex variable (Russian), Uspehi Matem. Nauk (N. S.), 7 (1952) no. 2(48), 31–122; English translation in Amer. Math. Soc. Translation 1954, (1954) no. 101, 99 pp.

    • Search Google Scholar
    • Export Citation
  • [17]

    Montgomery, H. L., Topics in Multiplicative Number Theory, Lecture Notes Math., vol. 227, Springer-Verlag, Berlin, Heidelberg, New York (1971).

  • [18]

    Montgomery, H. L., The pair correlation of zeros of the zeta function, in: Analytic Number Theory (ed. H. G. Diamond) (St. Louis Univ., 1972), Proc. Sympos. Pure Math., vol. XXIV, Amer. Math. Soc., Providence (1973), 181193.

    • Search Google Scholar
    • Export Citation
  • [19]

    Sander, J. and Steuding, J., Joint universality for sums and products of Dirichlet L-functions, Analysis, 26 no. 3 (2006), 295312.

  • [20]

    Sarason, D., Complex Function Theory, Amer. Math. Soc., Providence (2007).

  • [21]

    Steuding, J., Value-Distribution of L-Functions, Lecture Notes Math., vol. 1877, Springer, Berlin, Heidelberg, New York (2007).

  • [22]

    Steuding, J., The roots of the equation ζ(s) = a are uniformly distributed modulo one, in: Anal. Probab. Methods Number Theory (eds. A. Laurincikasˇ , TEV, Vilnius (2012), 243249.

    • Search Google Scholar
    • Export Citation
  • [23]

    Titchmarsh, E. C., The Theory of the Riemann zeta-Function, Second edition, Edited by D. R. Heath-Brown, Clarendon Press, Oxford (1986).

    • Search Google Scholar
    • Export Citation
  • [24]

    Voronin, S. M., A theorem on the“universality”of the Riemann zeta-function (Russian), Izv. Akad. Nauk SSSR, Ser. Matem., 39 (1975) no. 3, 475486; English translation in Math. USSR-Izv., 9 (1975) no. 3, 443–453.

    • Search Google Scholar
    • Export Citation