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 $\bar{s}$ 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 $\zeta (s)=\sum 1/n^s$.

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

This means $\zeta (\bar{s})$ 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 $\eta (\bar{s})$ 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 $1/\bar{z}=\overline{1/z}$.

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

Therefore we have:

$$\begin{array}{rl}\zeta (\bar{s})& =(1-2^{1-\bar{s}}{)}^{-1}\cdot \eta (\bar{s})\\ \\ & =\overline{(1-2^{1-s}{)}^{-1}}\cdot \overline{\eta (s)}\\ \\ & =\overline{\zeta (s)}\end{array}$$

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.