I have a very simple question about the Riemann Hypothesis that's probably quite obvious to somebody with more experience in complex analysis. I was having trouble with the same type of problem in my functional analysis book, as well.
Why does the truth of
$$ \frac{\zeta(s + \Delta r e ^{i\theta})}{||\zeta(s + \Delta r e ^{i\theta})||} = -\frac{\zeta(s - \Delta r e ^{i\theta})}{||\zeta(s - \Delta r e ^{i\theta})||} : \theta \in \mathbb{R}/2\pi\mathbb{Z}, \Delta r \rightarrow 0^{+} $$
imply that $\zeta(s) = 0$?