Abstract
In this paper, we define the Dirichlet series \( \zeta_{u_T j} (s)\), \( j = 1, \dots, r\), absolutely converging in the half-plane \( \operatorname{Re} s> 1/2 \) and prove that the set of shifts \( (\zeta_{u_T 1} (s + ia_1 \tau), \dots, \zeta_{u_T r} (s + ia_r \tau)) \) approximating a given set of analytic functions has a positive density on the interval \( [T, T + H]\), \( H = o (T) \) as \( T \to \infty\). Here \( a_1, \dots, a_r \in \mathbb{R} \) are algebraic numbers linearly independent over \( \mathbb{Q} \) and \( u_T \to \infty \) as \( T \to \infty\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. M. Voronin, “Theorem on the “universality” of the Riemann zeta-function,” Math. USSR-Izv. 9 (3), 443–453 (1975).
H. Bohr and R. Courant, “Neue Anwendungen der Theorie der Diophantischen Approximationen auf Riemannschen Zetafunktion,” Angew. Math. 144, 249–274 (1914).
A. Laurinčikas, Limit Theorems for the Riemann Zeta-Function (Kluwer Acad. Publ., Dordrecht, 1996).
Ł. Pańkowski, “Joint universality for dependent \(L\)-functions,” Ramanujan J. 45 (1), 181–195 (2018).
A. Laurinčikas, R. Macaitienė, and D. Šiaučiūnas, “A generalization of the Voronin theorem,” Lith. Math. J. 59 (2), 156–168 (2019).
A. Reich, “Werteverteilung von Zetafunktionen,” Arch. Math. 34, 440–451 (1980).
A. Dubickas and A. Laurinčikas, “Distribution modulo 1 and the discrete universality of the Riemann zeta-function,” Abh. Math. Semin. Univ. Hambg. 86 (1), 79–87 (2016).
R. Macaitienė, “On discrete universality of the Riemann zeta-function with respect to uniformly distributed shifts,” Arch. Math. 108 (3), 271–281 (2017).
A. Laurinčikas, “Discrete universality of the Riemann zeta-function and uniform distribution modulo 1,” St. Petersburg Math. J. 30 (1), 103–110 (2019).
R. Garunkštis, A. Laurinčikas, and R. Macaitienė, “Zeros of the Riemann zeta-function and its universality,” Acta Arith. 181 (2), 127–142 (2017).
M. Korolev and A. Laurinčikas, “A new application of the Gram points,” Aequationes Math. 93 (5), 859–873 (2019).
A. Laurinčikas, “On the universality of the Riemann zeta-function,” Lith. Math. J. 35 (4), 399–402 (1995).
A. Laurinčikas, D. Šiaučiūnas, and G. Vadeikis, “Weighted discrete universality of the Riemann zeta-function,” Math. Model. Anal. 25 (1), 21–36 (2020).
A. Laurinčikas and L. Meška, “Sharpening of the universality inequality,” Math. Notes 96 (6), 971–976 (2014).
S. M. Voronin, “On functional independence of Dirichlet \(L\)-functions,” Acta Arith. 27, 493–503 (1975).
A. Laurinčikas, “On joint universality of the Riemann zeta-function,” Math. Notes 110 (2), 210–220 (2021).
A. Laurinčikas, “Approximation of analytic functions by an absolutely convergent Dirichlet series,” Arch. Math. 117 (1), 53–63 (2021).
M. Jasas, A. Laurinčikas, and D. Šiaučiūnas, “On the approximation of analytic functions by shifts of an absolutely convergent Dirichlet series,” Math. Notes 109 (6), 876–883 (2021).
A. Laurinčikas and D. Šiaučiūnas, “Discrete approximation by a Dirichlet series connected to the Riemann zeta-function,” Mathematics 9 (10), 1073 (2021).
M. Jasas, A. Laurinčikas, M. Stoncelis, and D. Šiaučiūnas, “Discrete universality of absolutely convergent Dirichlet series,” Math. Model. Anal. (in press).
S. N. Mergelyan, “Uniform approximations of functions of a complex variable,” Uspekhi Mat. Nauk 7 (2 (48)), 31–122 (1952).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematicheskie Zametki, 2022, Vol. 111, pp. 15-23 https://doi.org/10.4213/mzm13222.
Rights and permissions
About this article
Cite this article
Garbaliauskienė, V., Šiaučiūnas, D. Joint Universality of Certain Dirichlet Series. Math Notes 111, 13–19 (2022). https://doi.org/10.1134/S0001434622010035
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434622010035