# E117: Prime-counting approximations: li(x) and friends

```{figure} ../_static/experiments/e117_hero.png
:alt: E117 hero
:class: experiment-hero

E117: Prime-counting approximations: li(x) and friends
```

**Tags:** `number-theory`, `model-checking`, `visualization`, `li-x`, `pnt`, `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

- Compare π(x) to li(x) and related approximations and visualize the signed error.

## 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/e117.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e117.json
:language: json
```

:::

## References

- See the relevant references in `refs.bib`.

## Related experiments

- {doc}`e123` (E123: π(x;q,a) vs. a simple baseline)
- {doc}`e061` (Chebyshev ψ(x) and prime powers)
- {doc}`e116` (E116: Gram points and zero-counting heuristics)
- {doc}`e058` (Divisor-count record highs)
- {doc}`e119` (E119: Summatory totient Φ(x) scaling check)