Hunt #76: the Riemann zeros in tuning units

Hunt #75 (chroma_hue/) closed the note-to-colour question and, in its aftermath, the operator asked for something bolder. The substitution t = 2 pi x / ln 2 turns the critical line into a tuning-quality score for the x-note equal temperament (Gene Ward Smith; Xenharmonic Wiki, "The Riemann zeta function and tuning"), and that community studies the peaks. This laboratory's subject is the zeros. Nobody seems to have asked what the zeros say in those units. Landau's formula answers it, and this hunt tests the answer on Odlyzko's tables. Write-up: docs/34-zeros-in-tuning-units.md.

id: zeta_temperament
question: In tuning units theta = gamma ln2 / 2pi, do the Riemann zeros avoid the equal temperaments that tune prime-power harmonics, with the deficit Landau's formula predicts, and ignore composite harmonics?
frontier: the Xenharmonic literature ranks equal temperaments by peaks of |zeta(1/2+it)| (OEIS A117536) and does not treat the zeros; Landau 1912 gives sum over zeros of n^{i gamma} = -(T/2pi) Lambda(n)/sqrt(n) + O(log T), which in these units is the Fourier coefficient of the zeros mod 1 at frequency log2 n
proposed_attack: compute the mod-1 Fourier coefficients of Odlyzko's first 100,000 zeros at every frequency log2 n for n up to 32, compare with -Lambda(n)/sqrt(n)/<log(gamma/2pi)>, and measure the smoothed zero density at integer, half-integer and fifths-perfect x against the prediction
dead_routes:
  - reading the peaks of |Z| as new (they are A117536 and the Xenharmonic Wiki, reproduced here only as calibration)
  - pair correlation of zeros mod 1 in tuning units at the 100,000-zero scale (signal below 0.01, not resolved)
required_oracles:
  - Odlyzko's zeros1 table under its pinned sha256, loaded through zeta.moments.load_odlyzko_zeros
  - the 2000 cached zeros in data/explicit_zeros.npz as an always-available second table
  - composite n as the built-in control (Lambda(n) = 0 must give a vanishing coefficient)
  - uniformly random points as the null for the size of each coefficient
kill_conditions:
  - a composite n shows a coefficient of the size a prime power shows
  - the deficit at integers fails to track 1/log(gamma/2pi) across height bands
  - the Odlyzko table fails its checksum or its declared row count
agents_may:
  - search
  - derive
  - code
  - attack
  - formalize
agents_may_not:
  - declare novelty
  - declare theorem status
  - promote their own claim

Files

Tests: tests/test_zeta_temperament.py. The Odlyzko table is fetched into data/odlyzko/zeros1 (gitignored) by the probe if absent, and refused if its sha256 is not 3436c916a7878261ac183fd7b9448c9a4736b8bbccf1356874a6ce1788541632.

Scope

Nothing here is evidence for RH (docs/08). Landau's formula is unconditional in the form used (the zeros are on the line in Odlyzko's range by verification, not by assumption). The hunt touches no core module.

Run manifest

id: zeta_temperament-2026-08-22-first
hunt: zeta_temperament
started: 2026-08-22T19:45-05:00
finished: 2026-08-22T20:40-05:00
ran:
  - .venv/bin/python hunts/zeta_temperament/probe_landau_edo.py
  - .venv/bin/python hunts/zeta_temperament/probe_peaks_edo.py
  - .venv/bin/python hunts/zeta_temperament/make_figures.py
  - .venv/bin/python -m pytest -q tests/test_zeta_temperament.py
outcome: on 100,000 zeros every prime-power frequency log2 n carries the coefficient Landau predicts to within 0.0012, every composite gives 0.0000, and the smoothed zero density at integer temperaments is 0.801 against 0.800 predicted
artifacts:
  - hunts/zeta_temperament/results_landau.json
  - hunts/zeta_temperament/results_peaks.json
  - figures/zeta_temperament_landau.png
  - figures/zeta_temperament_density.png
  - docs/34-zeros-in-tuning-units.md