Universality of the Hurwitz zeta-function in short intervals

In the paper, we consider approximation of analytic functions by shifts \(\zeta (s+i\tau , \alpha )\) $\zeta (s+i\tau ,\alpha )$ , \(s=\sigma +it\) $s=\sigma +it$ , of the Hurwitz zeta-function with transcendental parameter \(\alpha \) $\alpha$ , for \(\tau \in [T, T+H]\) $\tau \in [T,T+H]$ , where \(T^{27/82}\leqslant H\leqslant T^{1/2}\) $T^{27/82}⩽H⩽T^{1/2}$ . We obtain that the set of these shifts approximating a given analytic function on the strip \(1/2< \sigma <1\) $1/2<\sigma <1$ has a positive density. For the proof, a mean square estimate for \(\zeta (\sigma +it, \alpha )\) $\zeta (\sigma +it,\alpha )$ with \(1/2< \sigma \leqslant 7/12\) $1/2<\sigma ⩽7/12$ in the interval \([T, T+H]\) $[T,T+H]$ and a probabilistic limit theorem in the space of analytic functions are applied.