# E055: Mertens function walk

```{figure} ../_static/experiments/e055_hero.png
:width: 80%
:alt: Preview figure for E055
```

```{figure} ../_static/experiments/e055_hero_2.png
:width: 80%
:alt: Preview figure for E055
```

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

## Highlights

- Plot $M(x)$ and scaled variants such as $M(x)/\sqrt{x}$.
- Compare qualitative behavior to a simple random-walk heuristic (without overclaiming).

## Goal

Build intuition for summatory Möbius behavior and typical fluctuation scales.

## Background (quick refresher)

- {doc}`../background/mobius-and-mertens`
- {doc}`../background/average-orders-and-erdos-kac`

## Research question

What does $M(x)/\sqrt{x}$ look like numerically over moderate ranges?

## Method

- Compute $\mu(n)$ up to $N$ and cumulative sums $M(x)$.
- Plot $M(x)$ and $M(x)/\sqrt{x}$; optionally add moving-window statistics.

## How to run

- `make run EXP=e055`
- `uv run python -m mathxlab.experiments.e055`

## Outputs

This experiment follows the standard output contract:

- `out/e055/figures/` — generated figures (PNG)
- `out/e055/report.md` — short narrative report
- `out/e055/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/e055.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e055.json
:language: json
```

:::

## References

See {cite:t}`tenenbaum2015introanalyticprobabilisticnumbertheory`, {cite:t}`odlyzkoteriele1985disproofmertensconjecture`.

## Related experiments

- {doc}`e105` (E105: Mertens M(x): scaling views)
- {doc}`e054` (Squarefree density via Möbius)
- {doc}`e056` (Liouville vs. Möbius walks)
- {doc}`e095` (E095: Squarefree filter: ω(n)=Ω(n) when μ(n)≠0)
- {doc}`e121` (E121: Möbius inversion as convolution undo)