A DISCRETE VERSION OF THE MISHOU THEOREM RELATED TO PERIODIC ZETA-FUNCTIONS

In the paper, we consider simultaneous approximation of a pair of analytic functions by discrete shifts $\zeta_{u_N}(s+ikh_1; \ga)$ and $\zeta_{u_N}(s+ikh_2, \alpha; \gb)$ of the absolutely convergent Dirichlet series connected to the periodic zeta-function with multiplicative sequence $\ga$, and the periodic Hurwitz zeta-function, respectively. We suppose that $u_N\to\infty$ and $u_N\ll N^2$ as $N\to\infty$, and the set $\{(h_1\log p:\! p\in\! \PP), (h_2\log(m+\alpha): m\in \NN_0), 2\pi\}$ is linearly independent over $\QQ$.