Tridimensional display of the Riemann Zeta function inside (-50.0,+50.0)x(-50.0,+50.0) (bird's-eye view) [Visualisation tridimensionnelle de la fonction Zêta de Riemann dans (-50.0,+50.0)x(-50.0,+50.0) (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 logarithm of the modulus of the Riemann Zeta function as a surface in a tridimensional space (the two dimensions of the complex plane plus the logarithm of the modulus). The so-called "phase" of the Riemann Zeta function (its argument) is displayed as colors painting the surface; the [0, 2.pi] segment is mapped on the {Blue,Red,Magenta,Green,Cyan,Yellow,White} set. On this surface the unique pole (z=1) as well as some of the roots (even negative integer numbers and points on the x=1/2 line -the famous Riemann's conjecture-) can be seen.

Here are more pictures about the Zeta function:





(CMAP28 WWW site: this page was created on 06/25/1999 and last updated on 09/16/2017 11:52:48 -CEST-)



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