# E086: Hardy Z(t) near zeros

```{figure} ../_static/experiments/e086_hero.png
:alt: E086 hero
:class: experiment-hero

E086: Hardy Z(t) near zeros
```

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

## Highlights

- Focused numeric experiment with a single main figure.
- Parameters saved to `params.json` for reproducibility.
- Lightweight computation suitable for CI “slow” suite (small defaults).

## What is computed

- A parameterized numeric evaluation related to the Riemann zeta function.
- A visualization summarizing the main phenomenon for the chosen parameter range.

## Algorithm sketch

1. Build the numeric grid / sampling points.
2. Evaluate the target quantity with controlled truncation.
3. Render the figure and write a short report.

## Outputs

- `report.md` — short narrative summary
- `params.json` — experiment parameters snapshot
- `figures/fig_01_hardy_Z_near_zeros.png` — main figure

## Published run snapshot

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

```{include} ../reports/e086.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e086.json
:language: json
```

:::

## References

{cite:t}`titchmarsh1986, edwards1974`

## Related experiments

- {doc}`e115` (E115: Hardy Z: sign changes and zero bracketing)
- {doc}`e084` (E084: |ζ(1/2+it)| growth snapshots)
- {doc}`e114` (E114: ζ via η: stability map on the critical line)
- {doc}`e087` (E087: Gram points and spacing)
- {doc}`e088` (E088: Zero counting via Riemann–von Mangoldt)