Universality of the Riemann zeta-function in short intervals

Abstract

By the Voronin theorem, the set of shifts of the Riemann zeta-function ζ(s+iτ), s=σ+it, τ∈R, that approximate any given non-vanishing analytic function defined on {s∈C:12<σ<1} has a positive lower density. In the paper, it is proved that the same property of the above shifts remains valid in the intervals [T,T+H] with T1/3(log⁡T)26/15⩽H⩽T.