If You like a computational proof of that make use of
Zeta (in Mathematica):
Zeta[1 - s] == \[Pi]^(1/2 - s) Gamma[s/2]/
Gamma[(1 - s)/2] Zeta[s] // FullSimplify
This is published by G. F. Bernhard Riemann himself in his 1859 original paper. Look for example at Riemann Hypothesis and references there in.
$\zeta (1-s)=\frac{\pi ^{\frac{1}{2}-s} \zeta (s) \Gamma \left(\frac{s}{2}\right)}{\Gamma
\left(\frac{1-s}{2}\right)}$
Do two limits. One from $0$ to $\frac{1}{2}$, and the other from $1$ to $\frac{1}{2}$. Or do variable change and look at the identity given: $t=1-s$. This proves immediately the question.
Often this is written as
$\Gamma
\left(\frac{1-s}{2}\right) \zeta (1-s)=\pi ^{\frac{1}{2}-s} \zeta (s) \Gamma \left(\frac{s}{2}\right)$
to show the symmetry to the critical line with $Re(s)=\frac{1}{2}$.
Now know that for $Re(s)=1$ there is no zero and for $Re(s)=0$ there is no zero. For $s=1$ on the real axis there is a singularity of order $1$.
So in the critical stripe there can only exist zero for $Re(s)=\frac{1}{2}$ following from Riemann identity for the Riemann $\zeta$ function.
Graphical the statement at the borders of the critical stripe can be shown with (in Mathematica)
Plot[{Abs@Zeta[I z], Abs@Zeta[1 + I z],
Abs@Zeta[1 + I Im@ZetaZero[1]]}, {z, 0, 20}]
![Riemann Zeta on the borders of the critical stripe](https://cdn.statically.io/img/i.sstatic.net/RpMTJ.png)
The green line is the functions value of the $\zeta$ function on the border $1$ for the first Riemann $\zeta$ on the critical stripe $Re(s)=\frac{1}{2}$. On the critical stripe the Riemann $\zeta$ is symmetrical to the real axis. So this is the first positive and negative zero. On the lower border $Re(s)=0$ the Riemann $\zeta$ functions is divergent. That is not so on the upper border. There the function remains bounded between the given lower limit and a small divergence of the maxima if You accept Riemann's hypothesis.
An exemplary and fun look at Riemann $\zeta$ function is
Plot[{Re@Zeta[I Im@ZetaZero[1] + z], Im@Zeta[z + I Im@ZetaZero[1]],
Abs@Zeta[z + I Im@ZetaZero[1]]}, {z, 0, 1}]
![across the critical stripe at the first Riemann zeta zero](https://cdn.statically.io/img/i.sstatic.net/qQOiu.png)
This is dominated by the real part and both get negative on the lower boundary. Mind the given identity above holds for both together.
A contourplot shows this
![contourplot at the first Riemann zeta zero](https://cdn.statically.io/img/i.sstatic.net/VpxbL.jpg)
This gives a nice idea of how the Riemann $\zeta$ behaves at the first zero over the critical line. This is not so symmetric as expected.
A the twelves zero this looks
![Riemann zeta over the critical stripe at his zeros](https://cdn.statically.io/img/i.sstatic.net/RlZop.jpg)
And at the 120's
![Riemann zeta over the critical stripe at his zeros](https://cdn.statically.io/img/i.sstatic.net/TigKj.jpg)
This shows that the divergence on the lower border of the critical stripe dominates the behavior across the stripe very much.
This is very exemplary and just for fun. But is shows up how complicated the situations is. And it allows hypothesizing that every zero of the Riemann $\zeta$ is individual and the behavior of the critical stripe varies a lot.
The Riemann $\zeta$ is steady and infinitely often differentiable up to the singularity at $s=1$.
This is not considered chaos. The Riemann $\zeta$ is an L-function and behaves well.
There is only the Riemann hypothesis open, whether there are infinitely many zero on the critical line $Re(s)=\frac{1}{2}$.
It is possible to transform the average line approximating, the Mangoldt lambda function, into the stair plot of the number of positive primes, in Mathematica this is the PrimePi
function, with the Riemann $\zeta$ zeros so that the stairs are reproduced exactly. But this is an infinity process. For finite numbers of Riemann $\zeta$ zeros used, the approximation remains undulated with high deviation at the steps.
Mind this is a requirement. Not a proof. This on the other defines the need to prove that there are infinitely many Riemann $\zeta$ zeros on the critical line, and that the proof give here holds on the critical stripe as well so that there are no other zeros present. As mentioned in a remark to another answer to the question, there are many zeros numerically calculated, but calculation is not a proof. This makes on the other hand sure that up very high values the Riemann $\zeta$ zeros are enumerated and calculated.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/Me4eT.png)
These zeros can get very close together as this example shows:
$\zeta$" />
This is the 23999 and 24000's zero. They are already close together. They differ by $0.498732$. The Plot3D
surface of the absolute value shows small narrow dips toughing the plane $\zeta[s]==0$ is single points only. In this region the values on the lower border of the critical strip are very large and drop steeply before the critical stripe. On the upper border of the critical stripe the function is nearly flat and not wavy.
The shown pairing might get very important. If there is a tendency in this pairing that the values get closer and closer for larger values. The http://www.dtc.umn.edu/~odlyzko/doc/zeta.html study did an extra check for the separateness of the zeros. They did not find something spectacular by this.
The distribution that this study confirmed was a normal Gaussian. This does not inhibit such touching, and showed that the class of very small distances get more and more filled the larger the values are along the imaginary line.
ContourPlot[
Abs[Zeta[x + I y]], {x, 0, 1}, {y, Im[ZetaZero[24000]] - 0.75,
Im[ZetaZero[24000]] + 0.25}, PlotPoints -> 300,
ColorFunction -> "Aquamarine",
Contours -> {0.02, 0.03, 0.1, 0.2, 0.3, 0.4, 0.5}]
![countour plot of a paired zero of the Riemann zeta](https://cdn.statically.io/img/i.sstatic.net/1ZVc0.jpg)
The structure is superposed the repellance from the lower border of the critical line and a attraction of the contour lines for the absolute value. For this behavior both imaginary part and real part need to be zero. The attraction shows that the imaginary part dominates.
These dips for the zero point are very narrow and really points on the critical line. The closer the point is, the more are the contour lines circle. The disturbances are not as strong as the zero is.
But these are presumable transcendent zeros. The first derivative for example does not have much zero at the points where the Riemann $\zeta[s]$ has. Since the values of the derivatives tell about the multiplicity of the zero, these zeros are mostly "transcendent", L-function-type zeros of the infinitely many factors type, in the literature there is sometime used the attribute simple. This is in contrast to my use.
If the absolute function is still considered, but the negative values are used in between the consecutive zeros than this looks steady still:
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/0m9QU.png)
![real part and imaginary part](https://cdn.statically.io/img/i.sstatic.net/NpdXM.png)
The plot of the real part and imaginary part show that along the critical line between these two zeros both contribute to the absolute value.
From the distribution point of view the extreme value have to be taken and confirm the validity of the contribution. It is possible to show that for larger values the narrower pairs occur more often.
ListPlot[ListConvolve[{1, -1},
ZetaZero[#] & /@ (Range[2500] + 1000)] // Im // N]
![list plot of the distances between Riemann zeta zeros](https://cdn.statically.io/img/i.sstatic.net/jnRjb.png)
This show already the very narrow zeros occur already for lower values of the Riemann $\zeta[s]$.
ListPlot[ListConvolve[{1, -1},
ZetaZero[#] & /@ (Range[2500] + 100000)] // Im // N]
![list plot of the distances between Riemann zeta zeros](https://cdn.statically.io/img/i.sstatic.net/i1S9b.png)
The same sample but at very much higher indices for the Riemann $\zeta[s]$. The trend is clear. The are less bigger distances between following each other zeros and much more smaller ones.
But the smallest one has not gotten much smaller. It only a little bit smaller.
This trend does not hold steadily. There are exceptions that not outliers. The distribution for the smaller values of the indices is somewhat wavy in the median. That is a fact all over the critical line. So the second distribution picture is a lucky choice.
As already mentioned, a distribution is for probabilities and the possible is possible. The bigger the number of considered zeros the more the distribution becomes valid. That is a proof for small ensembles known to statistician very well.
For the packed array of $\zeta$ zeros in Mathematica the end shows even smaller distances between pairs of zeros:
.
I hope that this gave some insights into the delicate situation on the critical stripe and the critical for the Riemann $\zeta$. This shows up what enormous deed, the Riemann identity. Riemann came from the functional equation approach not from the heuristic, empirical perspective as these examples are, and the work of Odlyzhko is. So statistics is mathematics, and this is the exemplary part of this proof approach.
Perhaps many of these coinside for very large indices. Maybe there are real intervals between the zeros in which the absolute function remains zeros all the way. As shown in the contour plot, and with the plots on the critical line, and the difference convolution of the consequtive zeros, the zeros get some attraction for each other from an unknown source.