E094: ω(n) vs. Ω(n): Erdős–Kac normalization

E094 hero

E094: ω(n) vs. Ω(n): Erdős–Kac normalization

Tags: number-theory, quantitative-exploration, visualization, omega, arithmetic-functions, numerics

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 ω(n) and Ω(n) distributions and their Erdős–Kac normalizations.

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_omega_vs_bigomega_hist.png

  • params.json

  • report.md

Notes

  • Mean ω(n) ≈ 2.725, Var ω(n) ≈ 0.888

  • Mean Ω(n) ≈ 3.497, Var Ω(n) ≈ 3.080

  • log log N ≈ 2.502 (Erdős–Kac heuristic scale)

params.json (snapshot)
{
  "max_k": 12,
  "n_max": 200000
}

References

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