On a Generalization of Voronin's Theorem

Voronin's theorem states that the Riemann zeta-function $\zeta(s)$ is universal in the sense that all analytic functions that are defined and have no zeros on the right half of the critical strip can be approximated by the shifts $\zeta(s+i\tau)$, $\tau \in \mathbb{R}$. Some results on the approximation by the shifts $\zeta(s+i\varphi(\tau))$ with some function $\varphi(\tau)$ are also known. In this paper, it is established that an analytic function without zeros in the strip $1/2+1/(2\alpha)<\operatorname{Re} s<1$ can be approximated by the shifts $\zeta(s+i\log^\alpha \tau)$ with $\alpha >1$.