# E099: Jordan totients J_k: identities and ratios

```{figure} ../_static/experiments/e099_hero.png
:alt: E099 hero
:class: experiment-hero

E099: Jordan totients J_k: identities and ratios
```

**Tags:** `number-theory`, `model-checking`, `visualization`, `totient`, `arithmetic-functions`

## 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

- Compute J_k(n), verify J_1=φ, and visualize ratios J_k(n)/n^k.

## 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/e099.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e099.json
:language: json
```

:::

## References

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

## Related experiments

- {doc}`e060` (Jordan totients)
- {doc}`e101` (E101: Reduced residues modulo q: concrete structure)
- {doc}`e052` (Totient ratio landscape)
- {doc}`e053` (Inverse totient multiplicities)
- {doc}`e062` (Carmichael λ(n) vs. φ(n))