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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
.
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