# E119: Summatory totient Φ(x) scaling check

```{figure} ../_static/experiments/e119_hero.png
:alt: E119 hero
:class: experiment-hero

E119: Summatory totient Φ(x) scaling check
```

**Tags:** `number-theory`, `quantitative-exploration`, `visualization`, `totient`, `summatory`, `bounds`

## 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

- Plot Φ(x)=∑_{n≤x} φ(n) and compare to the main-term scaling 3x²/π².

## 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/e119.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e119.json
:language: json
```

:::

## References

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

## Related experiments

- {doc}`e105` (E105: Mertens M(x): scaling views)
- {doc}`e052` (Totient ratio landscape)
- {doc}`e053` (Inverse totient multiplicities)
- {doc}`e058` (Divisor-count record highs)
- {doc}`e117` (E117: Prime-counting approximations: li(x) and friends)