Riemann Zeta Function shows non-trival zeros critical-strip symmetry

[binbagsss]

1. Homework Statement

I want to show that the non-trival zeros of the Riemann Zeta function all lie in the critical strip $0 < Re(s) < 1$ and further to this that they are symmetric about the line $Re(s)= 1/2$

where $\zeta(s) = \sum\limits^{\infty}_{n=1}n^{-s}$

With the functional equation $\zeta(1-s)=\pi^{1/2 - s} \frac{\Gamma(s/2)}{\Gamma(1/2(1-s))} \zeta (s)$

2. Homework Equations

see above

3. The Attempt at a Solution

My first step is to narrow down to $Re(s) < 1$ since for $Re(s)>0$ the following holds:

$\zeta (s) = \Pi_{p} \frac{1}{1-p^{-s}} \neq 0$ since no terms are zero.

Then using the functional equation $\zeta(1-s)=\pi^{1/2 - s} \frac{\Gamma(s/2)}{\Gamma(1/2(1-s))} \zeta (s)$ and sending $s=-2k$ and making $\zeta(-2k)$ the subject, it is clear that the non-trival zeros occur at $s=-2k$ arising from the reciprocal of the poles of $\Gamma(s)$ for $s=-2,-4...$

I also have that $\zeta(s)$ has a pole at $s=1$ and $\zeta(0)=-1/2$ so I now have the critical strip as $0< Re(s) < 1$ .

I am now stuck what to do how to prove that they are symmetric about $s=1/2$. Any hints on getting started, help, greatly appreciated.

Many thanks

 

[fresh_42]

Without being in the subject, Wikipedia provides some hint concerning the symmetry at $Re(s)=\frac{1}{2}.$

Riemann found his conjecture while he investigated $\, \xi(s):=\pi^{-\frac{s}{2}}\,\Gamma(\frac{s}{2})\zeta(s)\,$ and observed $\xi(s)=\xi(1-s).$ Riemann himself used $s=\frac{1}{2}+it$ and got $\xi(\frac{1}{2}+it)=\xi(\frac{1}{2}-it)$ for all $t\in \mathbb{C}\,.$

I haven't checked any details, but it might help here. The link to the original "paper" is:
https://de.wikisource.org/wiki/Über_die_Anzahl_der_Primzahlen_unter_einer_gegebenen_Größe
(Wrong language, of course, but maybe you can read it anyway or think along the formulas.)