Joint value-distribution of shifts of the Riemann zeta-function

TL;DR

This paper demonstrates that the Riemann zeta-function's shifts can approximate any set of non-zero complex values, revealing new insights into its value distribution along certain lines.

Contribution

It establishes that for fixed complex s in the right half of the critical strip, the shifts of ζ(s) can approximate any non-zero complex values for infinitely many shifts τ.

Findings

Any non-zero complex values can be approximated by shifts of ζ(s) for infinitely many τ.

The result holds for fixed s in the right half of the critical strip.

The approximation applies to multiple values simultaneously with specified shifts.

Abstract

We prove that any non-zero complex values $z_1,\ldots,z_n$ can be approximated by the following integral shifts of the Riemann zeta-function $\zeta(s+id_1\tau),\ldots,\zeta(s+id_n\tau)$ for infinitely many $\tau$, provided $d_1,\ldots,d_n\in\mathbb{N}$ and $s$ is a fixed complex number lying in the right open half of the critical strip.