The Riemann zeta function ζ(s) extends the familiar sum 1 + 1/2ˢ + 1/3ˢ + … to the whole complex plane. Its "nontrivial" zeros all live inside the critical strip 0 < Re(s) < 1, and the Riemann Hypothesis conjectures every one of them sits exactly on the critical line Re(s) = 1/2. This scene traces ζ(σ + it) as a 3D curve: height is the imaginary part t, and the horizontal position is the complex value ζ(σ+it) itself (Re on one axis, Im on the other). The vertical line through the origin is the "zero axis" — whenever the curve swings back to touch it, that height t is a zero.
Over ten trillion nontrivial zeros have been computed and every single one sits exactly on the critical line — strong numerical evidence for the Riemann Hypothesis, still one of the seven Clay Millennium Prize Problems, unsolved after more than 160 years.
A 3D trace of ζ(σ+it) winding around the "zero axis" as t climbs, showing how the curve grazes zero repeatedly on the critical line σ=0.5 but never does at σ=1 — the fact behind the Prime Number Theorem.
Height represents the imaginary part t; horizontal position is the complex value ζ(σ+it) itself. Touching the vertical axis means ζ(σ+it)=0 at that height — this only recurs at σ=0.5.
Drag σ away from 0.5 and watch the curve pull off the axis; snap back to see it graze the marked zero heights. Toggle the σ=1 comparison curve and the zero-counting stats to connect zeros with prime distribution.
Over ten trillion nontrivial zeros have been computed and all sit precisely on the critical line — the Riemann Hypothesis remains unproven, one of the seven Clay Millennium Prize Problems.