# 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