Tridimensional display of the Riemann Zeta function inside [+0.1,+0.9]x[0,+50] [Visualisation tridimensionnelle de la fonction Zêta de Riemann dans la bande [+0.1,+0.9]x[0,+50]].

Here is the meaning of the three {X,Y,Z} display coordinates:

X = Re(Zeta(z))
Y = Im(Zeta(z))
Z = Re(z)

the complex number 'z' being defined inside [+0.1,+0.9]x[0,+50].

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|
```

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(CMAP28 WWW site: this page was created on 01/23/2012 and last updated on 03/04/2019 12:37:13 -CET-)

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