File:Voronin-3.png

DescriptionVoronin-3.png	English: The Riemann zeta-function on the strip 1/2<Re(s)<1, 103<Im(s)<109, to illustrate Voronin's universality theorem. Français : La fonction Zeta de Riemann dans la bande 1/2<Re(s)<1, 103<Im(s)<109, pour illustrer le théorème d'universalité de Voronin.
Date	December 2007
Source	self-made, with Mathematica 6.0   This diagram was created with Mathematica.
Author	AxelBoldt

The image uses the following color function: . This means that the color of a point s encodes the value ζ(s) in the following manner: the hue of the color encodes the argument of the value, and the strength of the color encodes its absolute value. Strong colors represent values close to zero (black=0), weak colors represent values far away (while=infinity).

The following Mathematica 6.0 code was used to generate the image:

ComplexGraph[f_, xmin_, xmax_, ymin_, ymax_, points_: 100] := 
 (* f is the complex function to be graphed in the region
 [xmin,xmax] * [ymin,ymax]. 
 The parameter points controls how many points will be
  sampled in each direction; good values are 100-500. *)
 RegionPlot[True, {x, xmin, xmax}, {y, ymin, ymax},
  ColorFunction -> Function[{x, y},
    Hue[Mod[Arg[f[x + I*y]], 2 Pi]/(2 Pi),
     1/(1 + 0.3 Log[Abs[f[x + I*y]] + 1]),
     1 - 1/(1.1 + 5 Log[Abs[f[x + I*y]] + 1])]],
  ColorFunctionScaling -> False,
  AspectRatio -> Automatic,
  PlotPoints -> points]

Show[ComplexGraph[Zeta, 1/2, 1, 103, 109, 200], AspectRatio -> 5]

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