Prime Overtones Lab
Two mirror-image pictures. In the first, every prime's divisibility wave is opened up into its own small chord — the fundamental at period p plus its overtones — and the chords interfere across the integers. In the second, the primes themselves are treated as overtones of one fundamental, so the waves shrink as the primes grow. The first is arithmetic and knows where the primes are. The second has forgotten divisibility, but something else comes out of it.
Coordinates to open up exact at integers
Overtone rows, and two column sums
The divisibility wave of k is exactly (1/k) Σr=0k−1 cos(2π r x / k). Each row here is one term: r = 0 is the constant 1/k, r = 1 is the fundamental with period k, and r = 2, 3, … are its overtones with periods k/2, k/3, …. Between integers the overtones wander; at every integer they interfere to exactly 1 when k | n and exactly 0 otherwise — which is why 2 needs two rows, 3 needs three, and p needs p. The rows r and k − r are the same cosine, so the merged view keeps one of each pair. The cyan strip adds every row: the count of selected coordinates dividing n, so primes above the bank are the silences. The green strip multiplies instead — "not 2, and not 3, and not 5 …" — and there the primes are the spikes.
Primes as overtones of one fundamental finite — a picture, not divisibility
The shrinking field, and its sum
Here prime p is the wave cos(2π p x / L): the pth overtone of a fundamental of length L. Nothing about "which integers does p divide" survives — but the sum is not noise. It spikes at the simple fractions of the frame, and the height and sign of each spike are set by the denominator: at x = L·a/q the primes' phases are spread evenly over the residues coprime to q, so the sum is about π(P) · μ(q) / φ(q) — positive at 1/6 and 5/6, negative at 1/2 and 1/3, near zero at 1/4 and 1/8, where μ is the Möbius function and φ Euler's totient. That is the primes seen from the other side of the Fourier transform: the "major arcs" of the circle method, the same structure Vinogradov used to prove that every large odd number is a sum of three primes. Push the prime limit up and the spikes sharpen. This view is a true fact about primes, but it is a fact about their residues, not their positions.
ζ(σ + it) as a sum of waves in t exact, up to the truncation shown
Rows: wave n has frequency ln n and amplitude n−σ. Strips: real part, imaginary part, and |sum|
This is the picture you were reaching for, with the two adjustments that make it the zeta function itself. ζ(s) = Σ n−s, and on the line s = σ + it each term is n−σ · (cos(t ln n) − i sin(t ln n)): a wave in t whose frequency is ln n (not n — the log squashes the shrinking: 2 → 0.69, 3 → 1.10, 109 → 4.69) and whose amplitude is n−σ (not equal — on the critical line that is 1/√n, and this exponent is the ½ of the Riemann hypothesis). Your "1/2, 1/3, 1/5" are these amplitudes at σ = 1, the harmonic series, which is where ζ has its pole. The real zeros are where all the waves cancel at once: the green strip touches the axis exactly at 14.13, 21.02, 25.01 … when σ = ½, and at no other σ. The plain partial sum needs the small tail term to be accurate for t below about 2πN; leave it on. The primes-only view is the Euler product written additively — it draws approximately log ζ, so it blows up (negatively) at the zeros instead of touching zero.