A discrete version of the Mishou theorem. II

In 2007, H. Mishou obtained a joint universality theorem for the Riemann zeta-function $\zeta (s)$ and the Hurwitz zeta-function $\zeta (s,\alpha )$ with transcendental parameter $\alpha $. The theorem states that a pair of analytic functions can be simultaneously approximated by the shifts $\zeta (s+i\tau )$ and $\zeta (s+i\tau ,\alpha )$, $\tau \in \mathbb R$. In 2015, E. Buivydas and the author established a version of this theorem in which the approximation is performed by the discrete shifts $\zeta (s+ikh)$ and $\zeta (s+ikh,\alpha )$, $h>0$, $k=0,1,2\dots {}\kern 1pt$. In the present study, we prove joint universality for the functions $\zeta (s)$ and $\zeta (s,\alpha )$ in the sense of approximation of a pair of analytic functions by the shifts $\zeta (s+ik^\beta h)$ and $\zeta (s+ik^\beta h,\alpha )$ with fixed $0<\beta <1$.