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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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up vote 1 down vote accepted

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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