Visualize ζ(s) in the critical strip 0 < Re(s) < 1. Color = phase, brightness = |ζ(s)|. Zeros on the critical line Re=1/2 appear as phase singularities (all colors meet at a point).
The Riemann Hypothesis (1859): all non-trivial zeros of ζ(s) lie on the critical line Re(s)=1/2. Known zeros: t₁≈14.13, t₂≈21.02, t₃≈25.01, ... The zero-free region controls the error in the Prime Number Theorem: π(x) ≈ Li(x), error ∝ xᵝ where β is the largest real part of a zero.