# E056: Liouville vs. Möbius walks

```{figure} ../_static/experiments/e056_hero.png
:width: 80%
:alt: Preview figure for E056
```

**Tags:** `number-theory`, `quantitative-exploration`, `visualization`, `arithmetic-functions`, `summatory`, `liouville`, `mobius`
See: {doc}`../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)

- {doc}`../background/liouville-function`
- {doc}`../background/mobius-and-mertens`
- {doc}`../background/omega-functions`

## 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).

```{include} ../reports/e056.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

::: {dropdown} params.json (snapshot)
:open:

```{literalinclude} ../params/e056.json
:language: json
```

:::

## References

See {cite:t}`tenenbaum2015introanalyticprobabilisticnumbertheory`.

## Related experiments

- {doc}`e054` (Squarefree density via Möbius)
- {doc}`e120` (E120: Liouville λ(n): partial sums and parity)
- {doc}`e055` (Mertens function walk)
- {doc}`e105` (E105: Mertens M(x): scaling views)
- {doc}`e092` (E092: 1/ζ(s) via the Möbius Dirichlet series)