Joint disctrete universality of Dirichlet $L$-functions. II

A. Laurinčikas ; D. Korsakienė ; D. Šiaučiūnas

Čebyševskij sbornik, Tome 16 (2015) no. 1, pp. 205-218

Résumé

In 1975, S. M. Voronin obtained the universality of Dirichlet $L$-functions $L(s,\chi)$, $s=\sigma+it$. This means that, for every compact $K$ of the strip $\{s\in \mathbb{C}: \tfrac{1}{2}\sigma1\}$, every continuous non-vanishing function on $K$ which is analytic in the interior of $K$ can be approximated uniformly on $K$ by shifts $L(s+i\tau,\chi)$, $\tau\in \mathbb{R}$. Also, S. M. Voronin investigating the functional independence of Dirichlet $L$-functions obtained the joint universality. In this case, a collection of analytic functions is approximated simultaneously by shifts $L(s+i\tau,\chi_1), \dots, L(s+i\tau,\chi_r)$, where $\chi_1,\dots,\chi_r$ are pairwise non-equivalent Dirichlet characters. The above universality is of continuous type. Also, a joint discrete universality for Dirichlet $L$-functions is known. In this case, a collection of analytic functions is approximated by discrete shifts $L(s+ikh,\chi_1), \dots, L(s+ikh,\chi_r)$, where $h>0$ is a fixed number and $k\in \mathbb{N}_0=\mathbb{N}\cup\{0\}$, and was proposed by B. Bagchi in 1981. For joint discrete universality of Dirichlet $L$-functions, a more general setting is possible. In [3], the approximation by shifts $L(s+ikh_1,\chi_1), \dots, L(s+ikh_r,\chi_r)$ with different $h_1>0,\dots, h_r>0$ was considered. This paper is devoted to approximation by shifts $L(s+ikh_1,\chi_1), \dots, L(s+ikh_{r_1},\chi_{r_1}), L(s+ikh,\chi_{r_1+1}), \dots, L(s+ikh,\chi_r)$, with different $h_1,\dots, h_{r_1}, h$. For this, the linear independence over $\mathbb{Q}$ of the set \begin{align*} L(h_1,\dots,h_{r_1}, h; \pi)=\big\{(h_1\log p:\; p\in \mathcal{P}), \dots, (h_{r_1}\log p:\; p\in \mathcal{P}),\\ (h\log p:\; p\in \mathcal{P});\pi \big\}, \end{align*} where $\mathcal{P}$ denotes the set of all prime numbers, is applied. Bibliography: 10 titles.

Détail

Keywords: analytic function, Dirichlet $L$-function, linear independence, universality.

Bibliographie

[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

[2] Billingsley P., Convergence of Probability Measures, Wiley, New York, 1968

[3] Dubickas A., Laurinčikas A., “Joint discrete universality of Dirichlet $L$-functions”, Archiv Math., 104 (2015), 25–35 | DOI

[4] Heyer H., Probability Measures on Locally Compact Groups, Springer-Verlag, Berlin–Heidelberg–New York, 1974

[5] Laurinčikas A., “On joint universality of Dirichlet $L$-functions”, Chebyshevskii Sb., 12:1 (2011), 129–139

[6] Mergelyan S. N., “Uniform approximations to functions of a complex variable”, Amer. Math. Trans., 101 (1954); Mergelyan S. N., “Ravnomernye priblizheniya funktsii kompleksnogo peremennogo”, UMN, 7:2 (1952), 31–122

[7] Montgomery H. L., Topics in Multiplicative Number Theory, Lecture Notes in Math., 227, Springer, Berlin, 1971 | DOI

[8] Steuding J., Value-Distribution of $L$-functions, Lecture Notes in Math., 1877, Springer-Verlag, Berlin–Heidelberg, 2007

[9] Voronin S. M., “Theorem on the “universality” of the Riemann zeta-function”, Math. USSR Izv., 9 (1975), 443–453 | DOI

[10] Voronin S. M., “The functional independence of Dirichlet $L$-functions”, Acta Arith., 27 (1975), 493–503