The Graph of the Riemann zeta function ζ(s)

The Riemann zeta function, in symbols ζ(s), is a rather complicated function which the mathematician Bernhard Riemann (18261866) introduced in 1859 to generalize the Euler zeta function
ζ(s)  = 
 n^{s}  (s > 1) 
to the complex plane. Since the Riemann zeta function ζ is a function from the complex plane ℂ to itself, i.e., ζ: ℂ → ℂ, its graph cannot be represented as a 3D image. Instead, the real part and the imaginary part are plotted separately. In addition, the absolute value ζ(s) may be shown.
You can download an executable Java program which plots the graph of the ζ function in this way:
http://mathit.org/Mathematik/Riemann/riemann.jar
(It requires Java JRE 6 or higher).
You can adjust the ranges of the plotted values Re s, Re s, and z, where s = Re s + i Im s, and z = Re ζ(s), z = Im ζ(s), or z = ζ(s), respectively.
In the plot, the real and imaginary lines are projected on the graph and are marked red. The socalled "critical line" (Re s = 1/2) is marked green; the famous Riemann hypothesis conjectures that all zeros besides s = 2, 4, 6, .... lie on the critical line.
If your browser has installed Java Applets, you can quickly start the program by simply clicking the button:
© de Vries 20042015 