Prime-well: shell measurements

What to measure

Loops of the ½ path — one loop between each pair of touches

Expected against found. A counting formula gives how many zeros a function has between two heights, across all shells together. If the ½ shell touches its south pole that many times, every zero in the range is on the ½ shell. Any shortfall is zeros sitting on other shells, two for each off-line pair. The formula is accurate to about one either way at each end of the range.

The ½ wave — the ½ shell alone, one number at each height

With the rotation undone, the ½ path only moves back and forth, so it can be drawn as a plain wave. The amber centre line is the south pole: every crossing is a zero, and every arch between two crossings is one loop. Its width is the gap and its height is the swing.

ζ

The look-alike

The zeros ring at the primes — the touches, played back as rhythms

Each prime gives ζ one wave, with a rhythm set by the prime. This chart runs that backwards: it takes only the heights where the ½ shell touches its south pole, treats each as a pure tone, adds them up, and asks which numbers they ring at. The amber ticks are the primes. More heights give sharper peaks.

ζ

The look-alike

The ½ shell on its sphere — one path, drawn for every height measured

Only the ½ shell, nothing else. With the rotation left in, each loop leaves the south pole in a different direction, so over many heights the one path paints the sphere. Tick the box to undo the rotation and the same path collapses onto a single circle. Drag to turn; double-click to reset.

Built from primes — the ½ wave assembled one prime at a time

ζ can be written as a product with one factor for each prime. This chart multiplies together only the first few factors and shows how far the result is from the south pole across the wave window above, next to ζ itself. Move the slider to add primes. The dips line up with ζ's zeros and deepen quickly over the first ten or so primes, then only creep. A product of finitely many primes never reaches the pole. The look-alike has no product like this, which is the ingredient it lacks.

The prime ladder

The same product for every number of primes at once: one prime on the bottom row, 300 on the top row, height across. Bright means close to the south pole. Each streak stands at one height and sharpens as primes are added, without moving. The ticks along the top are ζ's zeros; the amber line is the slider's position.

The prime globe

The ladder wrapped around a sphere and raised into relief. Going around the globe is the height t across the wave window; going from the bottom to the top is the number of primes, 1 to 300. The surface bulges where the product comes close to the south pole, so each of ζ's zeros is a ridge running up the globe, and the lumps on it come and go as primes are added. The amber ring is the slider's position. Drag to turn; double-click to reset.

How far each dip sits from its zero

One thin line for each of ζ's zeros in the window: how far the product's nearest dip is from that zero, in height, as primes are added. The amber line is the average distance. Lines that run flat along the middle are dips that have locked on to a zero and stopped moving.

Subtracting the ripple. The fine ripple and the dark bands come from stopping the product sharply at one prime. The primes left out can be allowed for in bulk, using only the smooth average rate at which primes occur, not their exact positions. With that correction the bands disappear, and the product keeps following ζ however many primes are used, instead of swinging away. What is left over is the difference between the actual primes and their smooth average, and that difference is exactly what the zeros encode.

What this does and doesn't show. Within these 300 primes the product follows ζ more and more closely. It does not keep improving forever on the ½ shell: once the primes used are much larger than about the square of the height, the plain product starts to swing away again. The product only settles to a fixed value on shells bigger than 1, where ζ has no zeros. On the ½ shell, ζ is defined by extending the function inward from there.

Every shell, measured — size σ across, one line per function

Distance map — how far every shell's path is from its south pole, at every height

Height runs across, shell size runs up. Bright means close to the south pole; a bright point is a zero. The amber tick marks σ = ½.

ζ

The look-alike

The glow in 3D — the distance map raised into a landscape

The same map as above, with closeness to the south pole as height: each glow becomes a peak, and a zero is a peak that reaches the top. The highlighted line is one shell's path across the landscape. Drag either picture to turn both; double-click to reset the view.

ζ

The look-alike

Order map — where a shell sits closer to zero than the shell inside it

Height runs across, shell size from ½ (bottom) to 1 (top). A lit cell means that shell is out of order at that height. An empty strip means every shell misses by more than the one inside it, at every height measured.

ζ

The look-alike

How to read all this. The loop measurements describe the ½ shell alone. The shell measurements compare shells of different sizes. The order map uses the values with the mirror rule's scaling included, which is the form in which "every shell stays in order, at every height" is the same statement as the Riemann Hypothesis. Everything here covers only the heights you choose; a clean result is a picture of the pattern over that range, not a proof that it continues.

Export what was measured

Each file covers both functions at the settings above. The CSV files open in any spreadsheet; the summary is a Markdown file for handing to another tool or collaborator.