Joint universality and generalized strong recurrence with rational parameter

Abstract

We prove that, for every rational $d\ne 0,\pm 1$ and every compact set $K\subset \{s\in \mathbb{C}:1/2<\Re (s)<1\}$ with connected complement, any analytic non-vanishing functions $f_1,f_2$ on $K$ can be approximated, uniformly on $K$, by the shifts $\zeta (s+i\tau )$ and $\zeta (s+id\tau )$, respectively. As a consequence we deduce that the set of $\tau$ satisfying $|\zeta (s+i\tau )-\zeta (s+id\tau )|<\epsilon$ uniformly on $K$ has a positive lower density for every $d\ne 0$.