E100: Carmichael λ(n) vs. Euler φ(n)

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E100: Carmichael λ(n) vs. Euler φ(n)

Tags: number-theory, quantitative-exploration, visualization, carmichael-lambda, 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

  • Compare λ(n) and φ(n) and visualize λ(n)/φ(n) over a finite range.

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_lambda_over_phi_hist.png

  • params.json

  • report.md

Notes

  • min λ(n)/φ(n): 0.002315

  • max λ(n)/φ(n): 1.000000

  • λ(n) is the exponent of (Z/nZ)^× and often much smaller than φ(n).

params.json (snapshot)
{
  "bins": 40,
  "max_ratio": 1.0,
  "n_max": 25000
}

References

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