Share:


On approximation of analytic functions by periodic Hurwitz zeta-functions

    Darius Siaučiūnas Affiliation
    ; Violeta Franckevič Affiliation
    ; Antanas Laurinčikas Affiliation

Abstract

The periodic Hurwitz zeta-function ζ(s, α; a), s = σ +it, with parameter 0 < α ≤ 1 and periodic sequence of complex numbers a = {am } is defined, for σ > 1, by series sum from m=0 to ∞ am / (m+α)s, and can be continued moromorphically to the whole complex plane. It is known that the function ζ(s, α; a) with transcendental or rational α is universal, i.e., its shifts ζ(s + iτ, α; a) approximate all analytic functions defined in the strip D = { s ∈ C : 1/2 < σ < 1. In the paper, it is proved that, for all 0 < α ≤ 1 and a, there exists a non-empty closed set Fα,a of analytic functions on D such that every function f ∈ Fα,a can be approximated by shifts ζ(s + iτ, α; a).


First Published Online: 21 Nov 2018

Keyword : Hurwitz zeta-function, periodic Hurwitz zeta-function, universality, weak con- vergence of probability measures

How to Cite
Siaučiūnas, D., Franckevič, V., & Laurinčikas, A. (2019). On approximation of analytic functions by periodic Hurwitz zeta-functions. Mathematical Modelling and Analysis, 24(1), 20-33. https://doi.org/10.3846/mma.2019.002
Published in Issue
Jan 1, 2019
Abstract Views
105
PDF Downloads
76
Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

References

P. Billingsley. Convergence of Probability Measures. Willey, New York, 1968.

A. Javtokas and A. Laurinčikas. On the periodic Hurwitz zeta-function. Hardy Ramanujan J., 29:18-36, 2006.

A. Javtokas and A. Laurinčikas. Universality of the periodic Hurwitz zetafunction. Integral Transforms Spec. Funct., 17:711-722, 2006.

A. Laurinčikas. Limit Theorems for the Riemann Zeta-Function. Kluwer, Dordrecht, 1996.

A. Laurinčikas, R. Macaitienė, D. Mochov and D. Šiaučiūnas. Universality of the periodic Hurwitz zeta-function with rational parameter. Sib. Math. J., 59(5):894-900, 2018.

K. Matsumoto. A survey on the theory of universality for zeta and L-functions.In M. Kaneko, S. Kanemitsu and J. Liu(Eds.), Number Theory: Plowing and Starring Through High Wawe Forms, Proc. 7th China-Japan Semin. (Fukuoka 2013), volume 11 of Number Theory and Appl., pp. 95-144, New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 2015. World Scientic Publishing Co.

S.N. Mergelyan. Uniform approximations to functions of complex variable. Usp. Mat. Nauk., 7:31-122, 1952 (in Russian).

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

S.M. Voronin. Theorem on the universality" of the Riemann zeta-function. Izv. Akad. Nauk SSSR, Ser. Matem., 39:475-486, 1975 (in Russian).