The Riemann Zeta Function in Four Dimensions

t = Im(s)

0.3

framing

on

off

The Riemann zeta function is the analytic continuation of the function

ζ(s)=

∞

Σ

n=1

1/

s

n

, where

s=σ+it∈

,

σ>1

. The blue curve is a plot of

u+iv=ζ(s)

. You can vary

t=Im(s)

using the slider; it acts as the fourth dimension. The black line marks the origin of the

u

-

v

complex plane. The red arrows mark where the zeta function (blue line) crosses the black line (

ζ=0

); these are some of the zeros of the zeta function. The so-called trivial zeros appear at negative even integers when

Im(s)=0

. The Riemann conjecture states that the nontrivial zeros all lie on the critical line

Re(s)=1/2

.