Oddity of a functional equation for the R zeta function

[nomadreid]

TL;DR

ζ(s)=ζ(1-s) for the zeta function seems to indicate a symmetry around Re(s)=1/2, but this is odd....

In https://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/fnleqn.htm the equation

ζ(s)=ζ(1-s) is used, where ζ is the Riemann zeta function, which I find curious, for the following reasons

this indicates a symmetry around Re(s)=1/2, which seems to be what the diagram at 20:27 of seems to imply, but contradicting the statement " The Riemann zeta function is not symmetric along with any vertical line at all " from https://www.quora.com/Is-the-Riemann-zeta-function-symmetrical

as well as the elementary consideration that on the real axis there are the trivial zeros in the negative reals that have no corresponding zeros in the positive-real-part side.

(Note that I am not asking about the symmetry ζ(s)=ζ(s*), which is more reasonable.)

What am I missing? Thanks in advance for your patience; I presume this question has been asked many times before (although I couldn't find a good answer with my Internet search).

 

[martinbn]

The functional equations is not for the zeta itself, but for the completed one.

$\zeta(s) = \sum_{n=1}^\infty \frac1{n^s}$

$\Lambda(s) = \frac12 \pi^{\frac{-s}2}s(s-1)\Gamma(\frac s2) \zeta(s)$

then

$\Lambda(s) = \Lambda(1-s)$

 

[fresh_42]

You can find the details (with formal proofs) in
https://www.physicsforums.com/insights/the-extended-riemann-hypothesis-and-ramanujans-sum/
and a bit of context in
https://www.physicsforums.com/insights/the-history-and-importance-of-the-riemann-hypothesis/.

 

[nomadreid]

Ah, super! Many thanks to both answers! martinbn's answer makes complete sense and clears my question about the equation up; fresh_42's links will provide valuable resources.

A side question, if I may: the 3Blue1Brown video (reference above) seems to be indicating a more complicated symmetry between the part for Re(s)>1 and Re(s)<1 -- his explanation is rather hand-wavy on that. That is, transforming the grid lines on each half of Re(s)=1, he graphs:

(How he graphs this is explained nicely in aheight's answer (post#2) in https://www.physicsforums.com/threads/video-analytic-continuation-seems-to-mix-4-d-2-d-maps.944596/ )

Although my question is rather broad, any hints that anyone could give about this symmetry would be highly appreciated. Either a quick equation as in martinbn's reply, or an indication as to where in the links provided by fresh_42 I might find the answer. Thanks again!

 

[fresh_42]

The symmetry is given by $\xi(s)=\xi(1-s)$ where $\xi(s)=\underbrace{\pi^{-s/2}\,\Gamma(s/2)}_{=c_s} \zeta(s).$

To get from the asymmetric $\zeta(s)$ to the symmetric $\xi(s)$ we need a correction factor $c_s$ which depends on the Gamma-function. And Euler has proven
$$\Gamma(z)\Gamma(1-z) = \dfrac{\pi}{\sin \pi z}\;\text{ for all }z\in \mathbb{C}-\mathbb{Z}$$
I suppose that "bending the grid lines" is a result of visualizing this property of the Gamma-function but I can only guess what exactly they did there. Maybe https://www.wolframalpha.com/input?i=y=pi/(Gamma(ix)*sin(pi+*+i*+x)) is an indication.

 

[nomadreid]

The symmetry is given by $\xi(s)=\xi(1-s)$ where $\xi(s)=\underbrace{\pi^{-s/2}\,\Gamma(s/2)}_{=c_s} \zeta(s).$

To get from the asyymetric $\zeta(s)$ to the symmetric $\xi(s)$ we need a correction factor $c_s$ which depends on the Gamma-function. And Euler has proven
$$\Gamma(z)\Gamma(1-z) = \dfrac{\pi}{\sin \pi z}\;\text{ for all }z\in \mathbb{C}-\mathbb{Z}$$
I suppose that "bending the grid lines" is a result of visualizing this property of the Gamma-function but I can only guess what exactly they did there. Maybe https://www.wolframalpha.com/input?i=y=pi/(Gamma(ix)*sin(pi+*+i*+x)) is an indication.

In the video he gives an example what he means by "bending the grid lines" by using the simpler transformation f(z)=z²: so he takes for example the grid line z: Im(z)=2i, so that if you square each point r+2i in the grid line (r real), you get (r+2i)²=(r²-4)+4ri, so that the line
{z| ∃r∈ℝ: z=(r²-4)+4ri is "bent" (and rotated) compared to the original line.
Roughly,blue to green:

So, it appears that the graph referred to in the last post would be looking at the zeta function as a transformation (or, a pair of transformations, one on either side of Re(s)=1) composed of rotations and scaling, so that between each z and ζ(z), a path is traced.