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:
                               \       -s
                    Zeta(s) =  /      n


                    \-/ s > 1

The complex Riemann Zeta function is defined as the serie:
                               \       -z
                    Zeta(z) =  /      n


                    \-/ 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:
                               \       -z
                    Zeta(z) =  /      n


                                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-)

[See all related pictures (including this one) [Voir toutes les images associées (incluant celle-ci)]]

[Please visit the related NumberTheory picture gallery [Visitez la galerie d'images NumberTheory associée]]
[Go back to AVirtualSpaceTimeTravelMachine [Retour à AVirtualSpaceTimeTravelMachine]]
[The Y2K bug [Le bug de l'an 2000]]

[Site Map, Help and Search [Plan du Site, Aide et Recherche]]
[Mail [Courrier]]
[About Pictures and Animations [A Propos des Images et des Animations]]

Copyright (c) Jean-François Colonna, 1999-2017.
Copyright (c) France Telecom R&D and CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 1999-2017.