# E078: Max partial sums across characters.

```{figure} ../_static/experiments/e078_hero.png
:width: 80%
:alt: Preview figure for E078
```

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

## Highlights
- Uses character orthogonality to reconstruct residue indicators.
- Measures reconstruction error / maximal partial sums across characters.
- Explores how primitive characters vary with the modulus.

## What this experiment does

For each Dirichlet character χ modulo q, define

The implementation focuses on a compact, reproducible numerical workflow: deterministic parameter defaults, structured output folders, and one or more figures saved for the gallery.

## Outputs

This experiment writes into `out/e078/`:

- `figures/fig_01_max_sums_hist.png`

## How to run

```bash
make run EXP=e078
```

## Notes

- The gallery preview figure shipped with the documentation uses conservative cutoffs so builds stay fast. If you run the experiment locally, increase the cutoffs to see the asymptotic regime more clearly.
- Prime-race plots depend on the chosen sampling of `x` (linear vs. log grid). The qualitative “who leads” picture can change when you zoom in.

## Published run snapshot

If this experiment is included in the docs gallery, include the published snapshot (report + params).

```{include} ../reports/e078.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e078.json
:language: json
```

:::

## References

{cite:t}`davenport2000multiplicativenumbertheory, nivenzuckermanmontgomery1991introductiontheorynumbers`

## Related experiments

- {doc}`e066` (Character partial sums: cancellation profiles.)
- {doc}`e110` (E110: Dirichlet L-series partial sums at s=1 and s=1/2)
- {doc}`e120` (E120: Liouville λ(n): partial sums and parity)
- {doc}`e065` (Orthogonality matrix for Dirichlet characters.)
- {doc}`e067` (Gauss sums: magnitude vs. sqrt(q).)