The Riemann Zeta Function in Four Dimensions
The Riemann Zeta Function in Four Dimensions
The Riemann zeta function is the analytic continuation of the function , where , . The blue curve is a plot of . You can vary using the slider; it acts as the fourth dimension. The black line marks the origin of the - complex plane. The red arrows mark where the zeta function (blue line) crosses the black line (); these are some of the zeros of the zeta function. The so-called trivial zeros appear at negative even integers when . The Riemann conjecture states that the nontrivial zeros all lie on the critical line .
ζ(s)=1/
∞
Σ
n=1
s
n
s=σ+it∈
σ>1
u+iv=ζ(s)
t=Im(s)
u
v
ζ=0
Im(s)=0
Re(s)=1/2