Voronin's universality theorem in short intervals

The famous Voronin universality theorem for the Riemann zeta-function $\zeta(s)$ says that a wide class of analytic functions can be approximated by shifts $\zeta(s+i\tau)$, $\tau\in \mathbb{R}$. More precisely, the set of shifts approximating a given analytic function has a positive lower density. We will discuss the density of the above shifts in short intervals, i.e., in $[T, T+H]$ with $H=o(T)$ as $T\to\infty$. Continuous and discrete cases will be considered. Moreover, approximation by generalized shifts $\zeta(s+i\varphi(k))$, with $\varphi(k)=\gamma_k$ and $\varphi(k)=t_k$, where $\{\gamma_k\}$ is a sequence of imaginary parts of non-trivial zeros of $\zeta(s)$, and $\{t_k\}$ is a sequence of the Gram points, will be discussed.

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