Joint Universality of Certain Dirichlet Series

V. Garbaliauskienė ; D. Siauciunas

Matematičeskie zametki, Tome 111 (2022) no. 1, pp. 15-23.

Résumé

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$.

Détail

Keywords: Riemann zeta function, Voronin's theorem, universality.

Bibliographie

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

[2] H. Bohr, R. Courant, “Neue Anwendungen der Theorie der Diophantischen Approximationen auf Riemannschen Zetafunktion”, Angew. Math., 144 (1914), 249–274 | Zbl

[3] A. Laurinčikas, Limit Theorems for the Riemann Zeta-Function, Kluwer Acad. Publ., Dordrecht, 1996 | MR

[4] Ł. Pańkowski, “Joint universality for dependent $L$-functions”, Ramanujan J., 45:1 (2018), 181–195 | DOI

[5] A. Laurinčikas, R. Macaitienė, D. Šiaučiūnas, “A generalization of the Voronin theorem”, Lith. Math. J., 59:2 (2019), 156–168 | DOI | Zbl

[6] A. Reich, “Werteverteilung von Zetafunktionen”, Arch. Math., 34 (1980), 440–451 | DOI | Zbl

[7] A. Dubickas, A. Laurinčikas, “Distribution modulo 1 and the discrete universality of the Riemann zeta-function”, Abh. Math. Semin. Univ. Hambg., 86:1 (2016), 79–87 | DOI | Zbl

[8] R. Macaitien{. e}, “On discrete universality of the Riemann zeta-function with respect to uniformly distributed shifts”, Arch. Math., 108:3 (2017), 271–281 | DOI | Zbl

[9] A. Laurinčikas, “Discrete universality of the Riemann zeta-function and uniform distribution modulo 1”, Algebra i analiz, 30:1 (2018), 139–150 | MR

[10] R. Garunkštis, A. Laurinčikas, R. Macaitienė, “Zeros of the Riemann zeta-function and its universality”, Acta Arith., 181:2 (2017), 127–142 | DOI | Zbl

[11] M. Korolev, A. Laurinčikas, “A new application of the Gram points”, Aequationes Math., 93:5 (2019), 859–873 | DOI | Zbl

[12] A. Laurinčikas, “On the universality of the Riemann zeta-function”, Lith. Math. J., 35:4 (1995), 399–402 | DOI | Zbl

[13] A. Laurinčikas, D. Šiaučiūnas, G. Vadeikis, “Weighted discrete universality of the Riemann zeta-function”, Math. Model. Anal., 25:1 (2020), 21–36 | DOI | Zbl

[14] A. Laurinchikas, L. Meshka, “Utochnenie neravenstva universalnosti”, Matem. zametki, 96:6 (2014), 905–910 | DOI | MR | Zbl

[15] S. M. Voronin, “O funktsionalnoi nezavisimosti $L$-funktsii Dirikhle”, Acta Arith., 27 (1975), 493–503 | DOI | Zbl

[16] A. Laurinchikas, “O sovmestnoi universalnosti dzeta-funktsii Rimana”, Matem. zametki, 110:2 (2021), 221–233 | DOI

[17] A. Laurinčikas, “Approximation of analytic functions by an absolutely convergent Dirichlet series”, Arch. Math., 117:1 (2021), 53–63 | DOI | Zbl

[18] M. Iasas, A. Laurinchikas, D. Shyauchyunas, “O priblizhenii analiticheskikh funktsii sdvigami absolyutno skhodyaschegosya ryada Dirikhle”, Matem. zametki, 109:6 (2021), 872–883 | DOI

[19] A. Laurinčikas, D. Šiaučiūnas, “Discrete approximation by a Dirichlet series connected to the Riemann zeta-function”, Mathematics, 9:10 (2021), 1073 | DOI

[20] M. Jasas, A. Laurinčikas, M. Stoncelis, D. Šiaučiūnas, “Discrete universality of absolutely convergent Dirichlet series”, Math. Model. Anal. (to appear)

[21] S. N. Mergelyan, “Ravnomernoe priblizhenie funktsii kompleksnogo peremennogo”, UMN, 7:2 (1952), 31–122 | MR | Zbl