# E063: Dirichlet convolution playground

```{figure} ../_static/experiments/e063_hero.png
:width: 80%
:alt: Preview figure for E063
```

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

## Highlights

- Compute Dirichlet convolutions numerically on a finite range.
- Check exact identities and report any mismatches (should be zero).

## Goal

Turn algebraic identities of arithmetic functions into concrete computed checks.

## Background (quick refresher)

- {doc}`../background/dirichlet-convolution`
- {doc}`../background/arithmetic-functions`
- {doc}`../background/euler-totient-function`

## Research question

Can we numerically verify standard Dirichlet-convolution identities up to a bound without mistakes?

## Method

- Implement divisor-sum convolution on [1..N].
- Compute μ*1 and μ*id and compare to ε and φ respectively.

## How to run

- `make run EXP=e063`
- `uv run python -m mathxlab.experiments.e063`

## Outputs

This experiment follows the standard output contract:

- `out/e063/figures/` — generated figures (PNG)
- `out/e063/report.md` — short narrative report
- `out/e063/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/e063.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e063.json
:language: json
```

:::

## References

See {cite:t}`apostol1976introanalyticnumbertheory`.

## Related experiments

- {doc}`e102` (E102: Dirichlet convolution identity zoo)
- {doc}`e121` (E121: Möbius inversion as convolution undo)
- {doc}`e111` (E111: Euler product vs. Dirichlet series for L(s,χ))
- {doc}`e055` (Mertens function walk)
- {doc}`e095` (E095: Squarefree filter: ω(n)=Ω(n) when μ(n)≠0)