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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