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Conference in memory of A. A. Karatsuba on number theory and applications, 2015
January 30, 2015 16:00–16:25, Moscow, Steklov Mathematical Institute of the Russian Academy of Sciences

A joint discrete universality of Dirichlet $L$-functions

A. Laurinčikas

Vilnius university, Department of Mathematics and Informatics, Vilnius
 Video records: Flash Video 160.8 Mb Flash Video 963.4 Mb MP4 160.8 Mb

Abstract: In 1975, S.M.Voronin, a student of professor A.A. Karatsuba, discovered a universality and joint universality of Dirichlet $L$-functions $L(s, \chi)$. Roughly speaking, the last means that a tuple of analytical functions can be approximated simultaneously by shifts $L(s+i\tau, \chi_1), ..., L(s+i\tau, \chi_r)$, $\tau\in {R}$. In 1981, B. Bagchi studied an approximation of a tuple of analytic functions by discrete shifts $L(s+ikh, \chi_1), ..., L(s+ikh, \chi_r)$, $k\in {N}_0=N\cup \{0\}$ with fixed $h>0$. In the talk, we consider a generalization of Bagchi's theorem on an approximation of analytical functions by discrete shifts $L(s+ikh_1, \chi_1), ..., L(s+ikh_r, \chi_r)$, $k\in {N}_0$, with fixed $h_1>0, ..., h_r>0$. Here the linear independence of the set
$$\{( h_1\log p: p\in {\cal P}) ..., ( h_r\log p: p\in {\cal P}); \pi\},$$
over the field of all rational numbers is needed (here $\cal P$ stands for the set of all prime numbers).