Geodesic
Discrete universality of the Riemann zeta-function and uniform distribution modulo~1
Algebra i analiz, Tome 30 (2018) no. 1, pp. 139-150.

Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that a wide class of analytic functions can be approximated by shifts $\zeta(s+i\varphi(k))$, $k\geqslant k_0$, $k\in\mathbb N$, of the Riemann zeta-function. Here the function $\varphi(t)$ has a continuous nonvanishing derivative on $[k_0,\infty)$ satisfying the estimate $\varphi(2t)\max_{t\leqslant u\leqslant2t}(\varphi'(u))^{-1}\ll t$, and the sequence $\{a\varphi(k)\colon k\geqslant k_0\}$ with every real $a\neq0$ is uniformly distributed modulo 1. Examples of $\varphi(t)$ are given.
Keywords: Riemann zeta-function, uniform distribution modulo 1, universality, weak convergence. 
@article{AA_2018_30_1_a5,
     author = {A. Laurin\v{c}ikas},
     title = {Discrete universality of the {Riemann} zeta-function and uniform distribution modulo~1},
     journal = {Algebra i analiz},
     pages = {139--150},
     publisher = {mathdoc},
     volume = {30},
     number = {1},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AA_2018_30_1_a5/}
}
TY  - JOUR
AU  - A. Laurinčikas
TI  - Discrete universality of the Riemann zeta-function and uniform distribution modulo~1
JO  - Algebra i analiz
PY  - 2018
SP  - 139
EP  - 150
VL  - 30
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2018_30_1_a5/
LA  - en
ID  - AA_2018_30_1_a5
ER  - 
%0 Journal Article
%A A. Laurinčikas
%T Discrete universality of the Riemann zeta-function and uniform distribution modulo~1
%J Algebra i analiz
%D 2018
%P 139-150
%V 30
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2018_30_1_a5/
%G en
%F AA_2018_30_1_a5
A. Laurinčikas. Discrete universality of the Riemann zeta-function and uniform distribution modulo~1. Algebra i analiz, Tome 30 (2018) no. 1, pp. 139-150. http://geodesic.mathdoc.fr/item/AA_2018_30_1_a5/

[1] Bagchi B., The statistical behaviour and universality properties of the Riemann zeta-function and other allied Dirichlet series, Ph. D. Thesis, Indian Stat. Inst., Calcutta, 1981

[2] Billingsley P., Convergence of probability measures, John Wiley Sons, Inc., New York, 1968 | MR

[3] 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 | DOI | MR

[4] Gonek S. M., Analytic properties of zeta and $L$-functions, Ph. D. Thesis, Univ. Michigan, 1979 | MR

[5] Kuipers L., Niederreiter H., Uniform distribution of sequences, Pure Appl. Math., Wiley-Intersci., New York, 1974 | MR

[6] Laurinčikas A., Limit theorems for the Riemann zeta-function, Math. Appl., 352, Kluwer. Acad. Publ., Dordrecht, 1996 | MR

[7] Mergelyan S. N., “Ravnomernye priblizheniya funktsii kompleksnogo peremennogo”, Uspekhi mat. nauk, 7:2 (1952), 31–122 | MR | Zbl

[8] Montgomery H. L, Topics in multiplicative number theory, Lecture Notes in Math., 227, Springer-Verlag, Berlin, 1971 | DOI | MR

[9] Reich A., “Werteverteilung von Zetafunktionen”, Arch. Math. (Basel), 34:5 (1980), 440–451 | DOI | MR

[10] Steuding J., Value-distribution of $L$-functions, Lecture Notes in Math., 1877, Springer, Berlin, 2007 | MR

[11] Voronin S. M., “Teorema ob “universalnosti” dzeta-funktsii Rimana”, Izv. AN SSSR. Ser. mat., 39:3 (1975), 475–486 | MR | Zbl

[12] Walsh J. L., Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc. Colloq. Publ., 20, Amer. Math. Soc., Providence, RI, 1965 | MR