The Riemann Zeta Function is Almost Symmetric

This blog is a quick note on the symmetry of the Riemann Zeta function.

The video for this blog is online [youtube], and the slides here [pdf].

The plots of the magnitude $|\zeta(s)|$ that we rendered previously suggest the function is symmetric about the real axis. This isn't quite true. 

Remembering the complex conjugate $\overline{s}$ is a reflection of s in the real axis, for example $\overline{3+2i}=3-2i$, let's look again at the terms in the series $\zeta(s)=\sum 1/n^s$.

$$n^{-\overline{s}}=e^{-\overline{s}\ln(n)}=\overline{e^{-sln(n)}}=\overline{n^{-s}}$$

This means $\zeta(\overline{s})$ is the complex conjugate of $\zeta(s)$. 

So, although the magnitude of $\zeta(s)$ is mirrored above and below the real axis, the sign of the imaginary part is inverted.

Let's also consider the recently developed series $\zeta(s)=(1-2^{1-s})^{-1}\eta(s)$. 

Following the same logic, we can say that $\eta(\overline{s})$ is the complex conjugate of $\eta(s)$, so this part has inverted phase above and below the real axis. 

We can say something similar for the other factor because $1/\overline{z}=\overline{1/z}$.

$$(1-2^{1-\overline{s}})^{-1} = \overline{\left(1-2^{1-s}\right)^{-1}}$$

Therefore we have:

$$\begin{align} \zeta(\overline{s}) &= ({1-2^{1-\overline{s}})^{-1}}\cdot\eta(\overline{s})\\ \\ &=\overline{(1-2^{1-s})^{-1}}\; \cdot \; \overline{\eta(s)} \\ \\ &= \overline{\zeta(s)} \end{align}$$

This gives us the same conclusions that the magnitude is mirrored in the real axis, but the phase is inverted.

The plot below shows the phase of $\zeta(s)$ coloured to illustrate this almost-symmetry.