# E054: Squarefree density via Möbius

```{figure} ../_static/experiments/e054_hero.png
:width: 80%
:alt: Preview figure for E054
```

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

## Highlights

- Use $\mu(n)^2$ as an indicator for squarefree integers.
- Plot the running density and compare to $6/\pi^2$.

## Goal

Show how an arithmetic function encodes a classic density result.

## Background (quick refresher)

- {doc}`../background/mobius-and-mertens`
- {doc}`../background/arithmetic-functions`

## Research question

How quickly does the empirical density of squarefree numbers approach $6/\pi^2$?

## Method

- Compute $\mu(n)$ up to $N$ and accumulate $\sum_{n\le x}\mu(n)^2$.
- Plot the ratio $\frac{1}{x}\sum_{n\le x}\mu(n)^2$ with a reference line at $6/\pi^2$.

## How to run

- `make run EXP=e054`
- `uv run python -m mathxlab.experiments.e054`

## Outputs

This experiment follows the standard output contract:

- `out/e054/figures/` — generated figures (PNG)
- `out/e054/report.md` — short narrative report
- `out/e054/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/e054.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e054.json
:language: json
```

:::

## References

See {cite:t}`apostol1976introanalyticnumbertheory`, {cite:t}`tenenbaum2015introanalyticprobabilisticnumbertheory`.

## Related experiments

- {doc}`e056` (Liouville vs. Möbius walks)
- {doc}`e055` (Mertens function walk)
- {doc}`e095` (E095: Squarefree filter: ω(n)=Ω(n) when μ(n)≠0)
- {doc}`e105` (E105: Mertens M(x): scaling views)
- {doc}`e092` (E092: 1/ζ(s) via the Möbius Dirichlet series)