# E090: Functional equation residual heatmap

```{figure} ../_static/experiments/e090_hero.png
:alt: E090 hero
:class: experiment-hero

E090: Functional equation residual heatmap
```

**Tags:** `analysis`, `quantitative-exploration`, `visualization`, `riemann-zeta`, `numerics`

## Highlights

- Focused numeric experiment with a single main figure.
- Parameters saved to `params.json` for reproducibility.
- Defaults are chosen, so the experiment remains feasible for the CI “slow” suite.

## What is computed

- Evaluate the functional-equation residual
  \(R(s)=\zeta(s)-\chi(s)\,\zeta(1-s)\)
  on a small rectangle on the complex plane.
- Visualize `log10(|R(s)| + eps)` as a heatmap.

## Notes

- This is a sanity-check style plot: away from poles/zeros and with sufficient precision, the residual should be small.
- Parameters control precision and grid size.

## Published run snapshot

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

```{include} ../reports/e090.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e090.json
:language: json
```

:::

## References

- See the zeta / Dirichlet-series references in `refs.bib`.

## Related experiments

- {doc}`e089` (E089: log|ζ(s)| heatmap)
- {doc}`e082` (E082: Zeta(s) series convergence)
- {doc}`e083` (E083: Series vs. Euler product (ζ))
- {doc}`e084` (E084: |ζ(1/2+it)| growth snapshots)
- {doc}`e085` (E085: Dirichlet eta acceleration for ζ(s))