A joint discrete universality of Dirichlet $L$-functions

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
$$ \left\{\left ( h_1\log p: p\in {\cal P}\right) ..., \left ( h_r\log p: p\in {\cal P}\right); \pi\right\}, $$
over the field of all rational numbers is needed (here $\cal P$ stands for the set of all prime numbers).