Build zeta one arrow at a time.
ζ(s) = Σ n⁻ˢ is usually handed to you as a finished object. Here it is manufactured in front of you: each term is an arrow, laid head to tail, and the point the walk lands on is the value. Drag s around the complex plane and watch convergence, the pole, and the zeros stop being definitions and start being things that visibly happen.
One HTML file, no build step, no network. The amber crosshair is a guard: ζ(s) computed independently by Euler–Maclaurin, so you can see the walk agree with it rather than take the drawing on faith.
Two channels
The input plane
Where are you standing?
Pick your s by dragging, with the critical strip drawn as an amber band, the ½ line as a dashed fault, the first zeros as cyan dots, and the pole at s = 1 marked with an ×. Drag near a zero and it snaps. East of the strip the raw series converges; inside it, never.
The walk
What does the sum actually do?
Every term becomes one arrow: length n⁻ᔆ, angle −t·ln n. Laid head to tail they walk the output plane. When the shrink beats the spin the walk lands, and where it lands is ζ(s). A zero is simply the walk that comes home to the origin.
Three moves, and the whole picture
Everything the forge does reduces to two knobs and one trick. Once those are physical rather than symbolic, most of the mystery around analytic continuation goes with them.
The shrink knob
The real part sets how fast each arrow shortens. Above 1 the shrink always wins and the walk converges. That is the whole of Euler's spring, and the reason it only flows east of the wall.
The spin knob
The imaginary part sets how fast each arrow rotates, against a logarithmic clock. It is the same log-clock the orchestra's waves read, which is why zeta's zeros turn into the primes' frequencies.
The bridge
Flip the signs to the alternating series and rescale by 1/(1 − 2¹⁻ˢ). The same terms now land inside the strip. Analytic continuation stops being a spell and becomes a move you can watch.
Forge it
- 1Start at s = 2, the Basel point. The walk spirals tight and lands on π²/6. Now slide toward the wall at re(s) = 1 and watch the landing get lazier until it stops landing at all.
- 2Jump to s = 1, the pole. The arrows stop shrinking fast enough to ever finish: the harmonic series, seen as a walk that escapes.
- 3Go to ½ + γ₁i and let all six hundred arrows out. The spiral closes back onto the origin. That is a zero of zeta, drawn rather than asserted. Then try ½ + 20i, on the line but not a zero, and watch it miss.
- 4Switch from the η bridge to the raw series inside the strip. Same terms, same s, and now the walk just drifts. The bridge is doing real work.
Three landmarks, made playable
The forge is the western end of the atlas, turned into something you can operate. Three of its landmarks stop being scenery here:
raw series = EULER SPRING · strip edge = CONVERGENCE WALL · η rescale = CONTINUATION BRIDGEThe spring is the walk landing while σ > 1. The wall is the exact place it stops landing. The bridge is the sign flip and rescale that carries the value across a wall the series cannot cross. That gap between what continues and what does not is the whole reason the Riemann Hypothesis is hard, and here you can watch it open up.
Once a zero is a closed spiral rather than a claim, the orchestra follows naturally: each of those zeros becomes one wave, and the waves add up to the primes.