# Shrinking circle on Zeta zeroes. $\zeta(s + \Delta r e^{i \theta})$ when $\zeta(s) = 0$ and $\Delta r \rightarrow 0^{+}$.

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 $\zeta(s) = 0$?

• $\zeta$ is continuous at $s$ so if $\zeta(s)\ne 0$ both sides approach a limit and we get $\zeta(s)/|\zeta(s)|=-\zeta(s)/|\zeta(s)|$ – saulspatz Mar 24 '18 at 17:32
• We are very far from RH here, only the most basic properties of holomorphic functions being involved. – Did Mar 24 '18 at 18:06