# E121: Möbius inversion as convolution undo

```{figure} ../_static/experiments/e121_hero.png
:alt: E121 hero
:class: experiment-hero

E121: Möbius inversion as convolution undo
```

**Tags:** `number-theory`, `model-checking`, `visualization`, `dirichlet-convolution`, `mobius`, `multiplicative`

## 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

- Recover a function from its Dirichlet convolution with 1 via Möbius inversion.

## 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/e121.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e121.json
:language: json
```

:::

## References

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

## Related experiments

- {doc}`e102` (E102: Dirichlet convolution identity zoo)
- {doc}`e063` (Dirichlet convolution playground)
- {doc}`e054` (Squarefree density via Möbius)
- {doc}`e056` (Liouville vs. Möbius walks)
- {doc}`e092` (E092: 1/ζ(s) via the Möbius Dirichlet series)