Graph of the Riemann zeta function ζ(s)

The Graph of the Riemann zeta function ζ(s)

The Riemann zeta function, in symbols ζ(s), is a rather complicated function which Bernhard Riemann (1826-1866) introduced in 1859 to generalize the Euler zeta function

ζ(s)

∞

n=1

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.

For a quick start of the program, simply click the button (requires Java VM 1.4 or higher):
You can download an executable jar-file (requires Java VM 1.4 or higher).

You can adjust the ranges of the plotted x, y, and z values, where s = x + iy, 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 so-called "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.