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




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:





(CMAP28 WWW site: this page was created on 03/17/2000 and last updated on 03/04/2019 12:36:41 -CET-)



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