ζ(s) = Σ n⁻ˢ, manufactured one term at a time · each step forges a complex number · drag s, watch the walk
CH·sINPUT PLANE · pick your s
presets
Vertical amber band = the critical strip; dashed fault = re(s) = ½; cyan dots = the
first zeros; × = the pole at s = 1. Drag anywhere (snaps to zeros when close). Green country
east of the strip: the raw series converges: Euler's spring flows. Inside the strip the raw
walk never settles: that boundary is the convergence wall, crossed below via the η bridge.
CH·ΣTHE WALK · partial sums in the output plane
seriessteps
walk open
Step n forges one arrow: length n⁻ᔆ (σ = the shrink rate, the envelope knob), spun by
−t·ln n (t = spin rate against the log-clock, the same clock the orchestra's waves read).
Head-to-tail, the arrows walk the output plane. σ > 1: shrink wins, the walk lands, and that
landing point is ζ(s). In the strip the raw walk drifts forever; flip the signs
(η(s) = Σ(−1)ⁿ⁻¹n⁻ˢ) and rescale by 1/(1−2¹⁻ˢ) and the same terms land: analytic
continuation as a playable move, not a spell. Amber crosshair = ζ(s) (Euler–Maclaurin guard,
computed independently of the walk). A zero is the walk that comes home: at ½ + γ₁i
the six-hundred-arrow spiral closes back onto the origin.