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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 $\mathbb{C}$?
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In the further notes at the end of Chapter XI to that reference (Titchmarsh "Theory of the Riemann Zeta Function")

"The problem of the distribution of values of $\zeta(1/2+it)$ is rather different from that of $\zeta(\sigma+i t)$ with $1/2<\sigma<1$. In the first place it is not know whether the values of $\zeta(1/2+i t)$ are everywhere dense, though one would conjecture so. Second, there is a difference in the rates of growth with respect to $t$..."

He then explains that for fixed $\sigma>1/2$, the values of $\log(\zeta(\sigma+i t))$ have a limiting distribution according to a theorem of Bohr and Jessen. On the other hand a theorem of Selberg states that the values of $\log(\zeta(1/2+it))$ when divided by $\sqrt{1/2\log(\log(t))}$, are distributed like a Gaussian.

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