# E060: Jordan totients

```{figure} ../_static/experiments/e060_hero.png
:width: 80%
:alt: Preview figure for E060
```

**Tags:** `number-theory`, `quantitative-exploration`, `visualization`, `arithmetic-functions`, `totient`, `multiplicative`
See: {doc}`../tags`.

## Highlights

- Compute $J_k(n)$ for k=1,2,3 and plot $J_k(n)/n^k$.
- Compare how the small-prime structure changes as k increases.

## Goal

Show a clean generalization of φ(n) and how normalization reveals multiplicative structure.

## Background (quick refresher)

- {doc}`../background/jordan-totient-function`
- {doc}`../background/euler-totient-function`

## Research question

How do the normalized Jordan totients $J_k(n)/n^k$ behave for k=1..3?

## Method

- Compute $J_k(n)=n^k\prod_{p\mid n}(1-p^{-k})$ via prime factors.
- Plot normalized values for k=1..3.

## How to run

- `make run EXP=e060`
- `uv run python -m mathxlab.experiments.e060`

## Outputs

This experiment follows the standard output contract:

- `out/e060/figures/` — generated figures (PNG)
- `out/e060/report.md` — short narrative report
- `out/e060/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/e060.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e060.json
:language: json
```

:::

## References

See {cite:t}`apostol1976introanalyticnumbertheory`.

## Related experiments

- {doc}`e099` (E099: Jordan totients J_k: identities and ratios)
- {doc}`e052` (Totient ratio landscape)
- {doc}`e102` (E102: Dirichlet convolution identity zoo)
- {doc}`e053` (Inverse totient multiplicities)
- {doc}`e062` (Carmichael λ(n) vs. φ(n))