# E122: Character averages over primes

```{figure} ../_static/experiments/e122_hero.png
:alt: E122 hero
:class: experiment-hero

E122: Character averages over primes
```

```{figure} ../_static/experiments/e122_hero_2.png
:alt: E122 hero
:class: experiment-hero

E122: Character averages over primes
```

```{figure} ../_static/experiments/e122_hero_3.png
:alt: E122 hero
:class: experiment-hero

E122: Character averages over primes
```

**Tags:** `number-theory`, `dirichlet-characters`, `l-functions`, `quantitative-exploration`, `visualization`, `aps`

## 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

- Average χ(p) over primes in arithmetic progressions and compared to equidistribution heuristics.

## 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/e122.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e122.json
:language: json
```

:::

## References

- See `docs/background/dirichlet-characters.md`.
- See `docs/background/dirichlet-l-functions.md` and `docs/background/dirichlet-convolution.md`.
- See `docs/background/primes-in-arithmetic-progressions.md` and `docs/background/prime-number-races.md`.

## Related experiments

- {doc}`e068` (Dirichlet L(s,χ): series vs. Euler product (partial approximations).)
- {doc}`e069` (L(1,χ): slow convergence and smoothing.)
- {doc}`e107` (E107: Conductor: primitive vs. induced characters)
- {doc}`e064` (Dirichlet character tables (phase view).)
- {doc}`e066` (Character partial sums: cancellation profiles.)