E099: Jordan totients J_k: identities and ratios

E099 hero

E099: Jordan totients J_k: identities and ratios

Tags: number-theory, model-checking, visualization, totient, arithmetic-functions

Highlights

  • Focused numeric experiment with a small set of figures.

  • Parameters saved to params.json for reproducibility.

  • Defaults are chosen, so the experiment remains feasible for the CI “slow” suite.

What is computed

  • Compute J_k(n), verify J_1=φ, and visualize ratios J_k(n)/n^k.

Notes

  • This page summarizes the intent; see the generated report.md in out/ for concrete outputs.

Published run snapshot

If this experiment is included in the docs gallery, include the published snapshot (report + params).

Artifacts

  • figures/fig_01_jordan_ratios.png

  • params.json

  • report.md

Notes

  • Mean ratios (downsampled) for k=1..k_max: [0.844324, 0.958415, 0.987113, 0.995815]

  • Max |J_1(n)-phi(n)| on a 100-point sample: 0

  • J_k(n) = n^k ∏_{p|n}(1-1/p^k) (multiplicative).

params.json (snapshot)
{
  "k_max": 4,
  "n_max": 30000,
  "stride": 10
}

References

  • See the arithmetic-functions background pages in docs/background/.