# E052: Totient ratio landscape

```{figure} ../_static/experiments/e052_hero.png
:width: 80%
:alt: Preview figure for E052
```

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

## Highlights

- Plot $\varphi(n)/n$ for a range of $n$ and annotate record lows.
- Relate deep dips to integers with many small prime factors (primorial-like structure).

  ## Goal

  Make the multiplicative structure of the totient function visible, especially the role of small primes.

  ## Background (quick refresher)

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

  ## Research question

  Which $n$ minimize $\varphi(n)/n$ up to a bound, and how do they relate to primorial products?

  ## Method

  - Compute $\varphi(n)$ for $n\le N$ (sieve / SPF-based).
- Plot the ratio $\varphi(n)/n$ and track record minima.
- Optionally compare record minima to primorials and nearby multiples.

  ## How to run

  - `make run EXP=e052`
  - `uv run python -m mathxlab.experiments.e052`

  ## Outputs

  This experiment follows the standard output contract:

  - `out/e052/figures/` — generated figures (PNG)
  - `out/e052/report.md` — short narrative report
  - `out/e052/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/e052.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e052.json
:language: json
```

:::

## References

  See {cite:t}`apostol1976introanalyticnumbertheory`, {cite:t}`tenenbaum2015introanalyticprobabilisticnumbertheory`, {cite:t}`nivenzuckermanmontgomery1991introductiontheorynumbers`.

## Related experiments

- {doc}`e060` (Jordan totients)
- {doc}`e053` (Inverse totient multiplicities)
- {doc}`e119` (E119: Summatory totient Φ(x) scaling check)
- {doc}`e102` (E102: Dirichlet convolution identity zoo)
- {doc}`e062` (Carmichael λ(n) vs. φ(n))