In Reference 4 of this Wikipedia article, it is stated that the curve $\{(\zeta(\sigma+it),\zeta^{(1)}(\sigma+it), \cdots, \zeta^{(n-1)}(\sigma+it))|t\in\mathbb R\}$ is dense in $\mathbb C^n$ if $\frac12 <\sigma < 1$.
- Is it known if the curve is still dense for some $\sigma\notin (\frac12,1)$?
- In particular, what if $\sigma=\frac12$?
- In particular, what if $\sigma=\frac12$ and $n=1$? i.e. is $\zeta(\frac12+it)$ dense in $\mathbb{C}$?