# E123: π(x;q,a) vs. a simple baseline

```{figure} ../_static/experiments/e123_hero.png
:alt: E123 hero
:class: experiment-hero

E123: π(x;q,a) vs. a simple baseline
```

**Tags:** `number-theory`, `prime-races`, `model-checking`, `visualization`, `aps`, `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;q,a) to a baseline x/(φ(q) log x) and visualize deviations.

## 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/e123.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e123.json
:language: json
```

:::

## References

- See `docs/background/primes-in-arithmetic-progressions.md` and `docs/background/prime-number-races.md`.

## Related experiments

- {doc}`e117` (E117: Prime-counting approximations: li(x) and friends)
- {doc}`e072` (Prime race mod 4: pi(x;4,3) vs. pi(x;4,1).)
- {doc}`e073` (Prime race mod 3: pi(x;3,2) vs. pi(x;3,1).)
- {doc}`e074` (Prime race mod 8: leaderboard among 1,3,5,7.)
- {doc}`e075` (Prime race statistic: distribution on a log-grid.)