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.