Sphere Shell Lab
Each size σ gets its own sphere, of radius σ, and the value of ζ (or of its look-alike) traces a path on it as the height climbs. The south pole is zero, so a zero is a touch at the south pole. Build the shells and step through them layer by layer, map which sizes ever touch, then measure the ½ shell's loops one by one and listen for the primes inside them. The last panel splits ζ into its starting loops, a piece and its mirror image, and tests where the two can cancel.
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.
The cylinder view. Choose "cylinders" beside the tick boxes and every shell is drawn as a cylinder of radius σ instead of a sphere. Going round the cylinder is the direction of the value. Going up or down is its size on a multiplying scale: one step up is always the same factor, the middle ring is size 1, the top is toward infinity and the bottom is toward zero. A turn of the path around the south pole on the sphere is one wrap around the cylinder here, so the stack of loops on a small shell is a path that spirals round and round. A zero is infinitely far down, so a touch shows as the path plunging off the bottom and coming back up half a turn further round. The picture is cut off a little above and below sizes 67 and 1⁄67, so very large and very small values sit on the top and bottom rims. Layers, flipping, growing, line strength and video export all work the same way in this view.
Video. Export video records the picture as it flips through the layers or grows from the inside, at the view, zoom and line strength you have set, and saves a video file to your downloads. The flip uses the "layers shown" setting (one at a time if it is set to all) and the "enlarge" tick box. The whole run is spread over the length you choose. Press the button again to stop early and keep what was recorded.
Beyond 0 to 1. Shells can run from just above 0 up to 12. Past σ = 1 the path shrinks toward a single point, the value 1, and what is left is a small circle of radius 1 ÷ 2σ drawn by the prime 2 alone, with ever fainter wobbles from 3, 5, 7 and so on.
Many shells. You can build up to 2000 shells at once. With many shells the lines are drawn thinner and fainter so the overall form shows through, only the first, last, ½ and 0.8085 shells are labelled, and the readout gives a summary. The line-strength slider makes every path fainter or bolder, which helps when hundreds of shells pile up. While you turn or zoom, a lighter sketch is drawn and the full picture returns when you stop. Grow from the inside adds the shells one after another, smallest first. Two thousand shells take a while to build and to redraw in full; if it is too slow, build fewer.
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.
The loops of the ½ shell · each loop runs from one touch of the south pole to the next
Uses the function and the heights chosen above. Click a loop in the chart or the table to light it up in white on the sphere.
What is measured. A loop is the stretch of the ½ shell's path from one touch of the south pole to the next, on a sphere of radius 1, without undoing the rotation. Rises to: the furthest the loop gets from the south pole, as an angle; 180° would be the north pole. Length around: the loop's circumference in degrees of arc; a full great circle is 360°. Area: the part of the sphere the loop encloses, as a share of the whole sphere. Total turning: how much the path steers to one side over the whole loop; a great circle has none. Opening at the pole: the angle between the direction the loop leaves the pole and the direction it comes back. Roundness: 1 for a perfect circle, less for anything stretched or dented. Turning, area and opening are tied together, so any two of them fix the third.
Are they all the same curve? If only wave 1 and its mirror image were present, every loop would be the same perfect circle (dashed): rising to 126.9°, 322.0° around, enclosing 27.6% of the sphere. The real loops are that circle bent by waves 2, 3, 4 and so on, and the bending differs from loop to loop. The opening at the pole averages 180°: a wider opening means a longer wait between touches, a narrower one a shorter wait. An opening beyond 360° means the path swung toward the pole and turned back short, so a pair of touches is missing from the ½ shell. That happens for the look-alike. For ζ it has never been found above the very lowest heights, and finding one would disprove the hypothesis.
The pitches hidden in the loops
Where the primes are. How far the path is from the south pole rises and falls as the height climbs: that is the loops' size. This panel splits that rise and fall into pure tones, using the logarithm of the distance so that doubling counts the same everywhere. For ζ the tones sit only at the primes and their powers, with strength 1 ÷ √p for a prime p, half of 1 ÷ p for its square, and so on (white rings), and there is silence at 6, 10, 12, 14 and 15. That is the product over the primes showing through on the ½ shell, even though the product itself does not settle there. The look-alike rings at different places, including composites such as 6, 14 and 21, because it is a blend of two functions and not a single product.
The starting loops · a piece, its mirror image, and where they can cancel
Uses the heights chosen at the top of the page, for ζ. Click the map to move the cross; the picture below it shows the loops at that shell and height.
The loops at the cross · the piece and its mirror, arrow by arrow
The loops. ζ is one loop for every whole number, added tip to tail. Loop n is an arrow of length 1 ÷ nσ that turns at speed log n as the height climbs, so a loop's place in the stack sets both its size and its speed. The piece is the first few of those loops added up. The picture of the loops can be moved: drag to pan, shift-drag or right-drag to turn it, scroll to zoom in on the small loops near the end of a chain, and double-click to reset. The mirror is the same loops reflected in the fold: on shell σ it uses the lengths they would have on shell 1 − σ, turned the other way, and scaled by the mirror rule's own factor. Piece plus mirror is a close copy of ζ when the piece has about as many loops as the height calls for, which is the square root of (height ÷ 2π).
The starting condition. A touch needs the piece and its mirror to cancel, and cancelling needs equal sizes. On the fold they are always equal in size. If the piece is bigger than its mirror on every shell bigger than ½ (the whole right half of the map teal), then piece plus mirror can touch only on the fold: that is the theorem. The map tests the condition for the loops you choose, and separately hunts for every touch of piece plus mirror. The condition is enough to force the touches onto the fold, but it is not required: a piece can fail it in places and still have all its touches on the fold. With a single loop, or the loops 1 and 2, the condition holds at every height shown. With more loops than the height calls for it fails, and touches appear off the fold in mirrored pairs.