Prime Well Lab

Your well

What you're seeing. The faint coloured waves are the individual waves; the bright white one is the Full System Well — all of them added by height. The walls are fixed, so every wave starts and ends at zero. Each wave bounces back and forth as a standing wave: it gives its energy once, and nothing builds up, because a standing wave is already the finished result of endless bouncing. With every whole number and equal energy, the sum becomes a straight ramp (the dashed line): the composite waves fill in exactly what the prime waves leave out, and all the individuality vanishes. Starting from wave 2 instead of wave 1, it is that ramp with wave 1 taken away — the gentle bend you see. Full cycles or half-cycles. A string's natural notes come in half-cycles: a single arch (the fundamental), then two half-cycles, three, and so on. "Wave n = n full cycles" uses only the even notes (2n half-cycles), so every wave is silent at the middle. "Wave n = n half-cycles" uses the string's own harmonic series: wave 2 is silent at the middle while wave 3 peaks there, and every odd wave pushes the middle up or down according to its remainder when divided by 4. This choice changes only this well — not the zeros, which come from the log-spaced recipe further down. With the prime waves alone the shape turns irregular, and its sharpest features sit at simple fractions of the well — the same meeting points as the bounce game.

What the well rings at — each wave's pitch and energy

A string between straight walls can only ring at whole-number multiples of its base note. Wave n, with n full cycles, rings at exactly n times the note of a single full cycle — so the pitches are 2, 3, 5, 7 … (or 4, 6, 8 … for composites): whole numbers. Which waves you put in decides how loud each of those notes is, never where the notes are. That is why the whole pattern returns exactly to its starting shape on every beat of that single-cycle note — watch the animation — and why this well, on its own, cannot ring at the zeros.

The switch — log spacing: the same well, turned into the zeros' own wave

What changed. Two settings: wave n now turns at speed log n as the height t climbs, instead of having n cycles across the well — and its height is 1 ÷ √n. Add the steady turning of the phase (the same sphere factor as in the sphere lab), and the Full System Well's combined wave is Riemann's own function on the ½ line (this is the Riemann–Siegel formula). The zeros are where it falls completely silent: every wave cancelling at once. With every setting at its default, its silences land on every zero in the window. What each switch teaches: drop the wave for 1, keep only the primes, or change the heights, and the silences leave the zeros — every whole number, 1 included, has to take part with exactly this height. And the number of waves taking part grows with the height: at t = 100 only waves 1, 2 and 3 are in play; wave 4 joins at t ≈ 100.5, and by t = 1,000 there are 12. The well adds waves as it climbs — the "more room as the pitch rises" that the thickening test asks for. This is a faithful way to compute the zeros, not a proof that they sit on the line: the recipe works on the line itself, and says nothing yet about why no zero lies off it.

Off the ½ line — can the wave be silent anywhere else? Both of its parts would have to be zero at the same instant

What "silent" means here. At each height t the combined wave has one value, and off the line that value has two parts — call them part A (cyan) and part B (amber), like two waves at right angles. The wave is silent at a height only if both parts are exactly zero there, at the same instant; that single point would be a zero. Cyan dots mark where part A crosses zero, amber dots where part B does, and a white ring marks a complete silence. The lower panel is the loudness — the size of the whole value, on a log scale — so a complete silence is a spike straight down to nothing. On σ = ½, part B vanishes everywhere (the mirror makes the wave a single real wave), so every crossing of part A is a complete silence: those are the zeros. Off the line, both parts wiggle and each crosses zero often, but for ζ their crossings never land on the same instant, so the loudness never reaches zero. The Riemann Hypothesis is the statement that this holds at every σ off the line, at every height, forever. The look-alike shows why it is not automatic: at σ = 0.808517 its two parts cross zero at the same instant near t = 85.7 — a complete silence off the line.

In 3D — the wave as a line spiralling around the height axis · a zero is where the line touches the axis

The spiral line. The height t runs along the white axis. Around it, the wave's two parts: part A up and down, part B toward and away from you. As t climbs, the point (A, B) swings around the axis, so the wave traces a line spiralling around it. A zero is where the line touches the axis — both parts zero at the same instant (white rings; violet ticks on the axis mark ζ's zeros on the line). Side — part A up is exactly the chart above; turn it and part B appears. Down the axis looks straight along the height, so you see the two parts face-on: the line loops around the centre, and a zero is a pass straight through it. On σ = ½ the line lies flat in one plane and passes through the axis at every zero; off the line it winds around the axis and, for ζ, never touches it. The look-alike's line touches the axis once off the line, at t = 85.7. The landscape. The floor is the input plane — σ across, height t going away — and the ground is raised by the loudness (log scale). Zeros are holes reaching the floor: for ζ they sit in a row along the amber ½ line; the look-alike has one off to the side. Drag to turn; shift-drag, right-drag, or the Drag to button to pan; scroll to zoom; double-click to reset. On a touchscreen, two fingers pan and pinch.

Your sphere-size idea — every σ as its own sphere, radius σ · a zero is where a shell's path touches its south pole

How it's built. Each shell is the Riemann sphere for one σ, shrunk to radius σ: the south pole is 0, the north pole is infinity, and the path is ζ's value as the height t climbs. All the south poles sit on one downward axis (white), so a zero is the moment a path touches the axis at its own shell's south pole (white ring). The margin map is the real test: for every size σ from 0 to 1 it records how close that shell's path ever comes to its south pole over the chosen heights, as an angle. A margin of 0° means the shell has a zero. For ζ the margin touches zero only at σ = ½; for the look-alike it also touches zero at 0.8085 and at its mirror, 0.1915. Mapping σ to size is a new way to picture the same question the off-line panel asks, not new information — but it makes the answer easy to see.

Looking at the layers. Choose how many layers to show at once, then step through them or let them flip on their own. Colours stay the same as in the full picture, and so do the sizes unless you tick the box that enlarges the shown layers. The small ring on a shown path marks its closest approach to its own south pole. With two or more layers shown, the readout also says how far apart neighbouring layers' paths are at the same height, which measures how much the path changes from one layer to the next. Within a fixed range of heights that change is limited, so a layer that clears its pole by a wide margin guarantees that the layers close to it also clear theirs. It is not a guarantee at every height: the margins shrink and the paths change faster as the height grows, and the look-alike's layers change just as smoothly while one of them touches.