E119: Summatory totient Φ(x) scaling check

E119 hero

E119: Summatory totient Φ(x) scaling check

Tags: number-theory, quantitative-exploration, visualization, totient, summatory, bounds

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

  • Plot Φ(x)=∑_{n≤x} φ(n) and compare to the main-term scaling 3x²/π².

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_psi_minus_x_smoothed.png

  • params.json

  • report.md

Notes

  • max |ψ(x)−x| (raw): 360.3

  • max |ψ(x)−x| (smoothed): 331.0

  • Smoothing suppresses prime-power spikes and reveals slower oscillations.

params.json (snapshot)
{
  "n_max": 300000,
  "stride": 10,
  "window": 500
}

References

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