E056: Liouville vs. Möbius walks

Preview figure for E056

Tags: number-theory, quantitative-exploration, visualization, arithmetic-functions, summatory, liouville, mobius See: Valid Tags.

Highlights

  • Plot partial sums of \(\lambda(n)\) and \(\mu(n)\) side by side.

  • Compare scaled versions to see how the inclusion/exclusion of squareful terms changes the walk.

Goal

Contrast two closely related ±1/0 sequences arising from prime factorizations.

Background (quick refresher)

Research question

How do the fluctuations of \(\sum_{n\le x}\lambda(n)\) compare to \(\sum_{n\le x}\mu(n)\)?

Method

  • Compute \(\Omega(n)\), then \(\lambda(n)=(-1)^{\Omega(n)}\); compute \(\mu(n)\).

  • Plot partial sums and scaled partial sums (e.g., divide by \(\sqrt{x}\)).

How to run

  • make run EXP=e056

  • uv run python -m mathxlab.experiments.e056

Outputs

This experiment follows the standard output contract:

  • out/e056/figures/ — generated figures (PNG)

  • out/e056/report.md — short narrative report

  • out/e056/manifest.json — snapshot metadata for the gallery

Published run snapshot

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

  • n_max: 300000

  • M(n_max) = 220

  • L(n_max) = -98

Figure:

  • fig_01_walks.png

params.json (snapshot)
{
  "n_max": 300000
}

References

See Tenenbaum [2015].