Tridimensional display of the Z=Zeta(Z) iteration inside [-20.0,+20.0]x[-20.0,+20.0] [Visualisation tridimensionnelle de l'itération Z=Zeta(Z) dans [-20.0,+20.0]x[-20.0,+20.0]].

The real Zeta function is defined as the serie:
```                            n=+infinity
_______
\
\       -s
Zeta(s) =  /      n
/______

n=1

\-/ s > 1
```

The complex Riemann Zeta function is defined as the serie:
```                            n=+infinity
_______
\
\       -z
Zeta(z) =  /      n
/______

n=1

\-/ z : Re(z) > 1
```

or again (Leonhard Euler):
```                              _________
|     |
|     |      1
Zeta(z) =  |     |  ---------
|     |        -z
|     |   1 - p

p E P
```
where 'P' denotes the set of the prime numbers 'p'.

It can be computed for all z with the following analytic continuation:
```                               n=N-1
_______
\
\       -z
Zeta(z) =  /      n
/______

n=1

1-z      -z
N        N
+ ------ + -----
z-1       2

k=V                        p=2k-2
_______                     ________
\          B                 |    |
\          2k    -z-(2k)+1  |    |
+  /      [-------.N           |    | (z+p)]
/______   (2k)!              |    |

k=1                          p=0

+ epsilon(z,N,V)

\-/ z : Re(z+2V+1) > 1

N ~ |z|
```

This picture displays the iteration of the Riemann Zeta function:
```                    Z  = C (current point)
0

Z  = Zeta(Z   )
n         n-1
```

(for each Z(0) inside a subset of the complex plane) as a surface in a tridimensional space (the two dimensions of the complex plane plus the number of iterations). The argument of the last computed Z(n) is displayed as colors painting the surface; the [0, 2.pi] segment is mapped on the {Blue,Red,Magenta,Green,Cyan,Yellow,White} set.

Here are more pictures about the iteration of the Zeta function:

• the number of irerations alone,
• the argument alone.

(CMAP28 WWW site: this page was created on 03/17/2000 and last updated on 02/08/2022 20:50:39 -CET-)

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