Universality of the zeta function in short intervals

Abstract

We improve the universality theorem of the Riemann zeta-function in short intervals by establishing universality for significantly shorter intervals $[T,T+H]$. Assuming the Riemann Hypothesis, we prove that universality in such short intervals holds for $H=(\log T{)}^B$ with an explicitly given $B>0$. Unconditionally, we show that for the same $H$ the set of real numbers $\tau \in [T,T+H]$ such that $\zeta (s+i\tau )$ approximates an arbitrary given analytic function has a positive upper density.

7 pages