# E105: Mertens M(x): scaling views

```{figure} ../_static/experiments/e105_hero.png
:alt: E105 hero
:class: experiment-hero

E105: Mertens M(x): scaling views
```

```{figure} ../_static/experiments/e105_hero_2.png
:alt: E105 hero
:class: experiment-hero

E105: Mertens M(x): scaling views
```

**Tags:** `number-theory`, `quantitative-exploration`, `visualization`, `mobius`, `summatory`, `open-problems`

## 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 M(x)=∑_{n≤x} μ(n) and rescalings such as M(x)/√x.

## 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).

```{include} ../reports/e105.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e105.json
:language: json
```

:::

## References

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

## Related experiments

- {doc}`e055` (Mertens function walk)
- {doc}`e054` (Squarefree density via Möbius)
- {doc}`e056` (Liouville vs. Möbius walks)
- {doc}`e119` (E119: Summatory totient Φ(x) scaling check)
- {doc}`e061` (Chebyshev ψ(x) and prime powers)