Riemann Hypothesis

Re(s)

0

range

20

The Riemann hypothesis is one of today's most important problems in mathematics. The hypothesis states that all of the nontrivial zeros of the Riemann zeta function are located on the critical line

Re(s)=1/2

. A $1,000,000 prize has been offered by the Clay Mathematics Institute for the first correct proof of the hypothesis.

The hypothesis was first formulated by Riemann in 1859 and has remained unsolved since then. It is known that the nontrivial zeros are located in the crtical strip

0<Re(s)<1

, moreover if we define

ξ(s)=

-s/2

π

Γ

s

2

ζ(s)

, then

ξ(s)=ξ(1-s)

, which shows that the zeros must be symmetric with respect to the critical line.

This Demonstration plots the absolute value of the zeta function with respect to the imaginary part of its argument. You can change the range of the plot and the real part of

s

. The dashed red lines show the position of the imaginary part of the zeros.