Joint universality and generalized strong recurrence with rational parameter

TL;DR

This paper establishes joint universality and strong recurrence properties of the Riemann zeta function with rational parameters, showing that shifts of the zeta function can approximate non-vanishing analytic functions uniformly.

Contribution

It proves joint universality and generalized strong recurrence for the Riemann zeta function with rational parameters, extending previous universality results.

Findings

Shifts of the zeta function can approximate any non-vanishing analytic functions on certain compact sets.

The set of shifts where the zeta function approximates itself with a rational shift has positive lower density.

Joint universality holds for rational parameters not equal to 0, ±1.

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 $|\zeta(s+i\tau)-\zeta(s+id\tau)|<\varepsilon$ uniformly on $K$ has a positive lower density for every $d\ne 0$.