# E091: Partial Euler products on the critical line

```{figure} ../_static/experiments/e091_hero.png
:alt: E091 hero
:class: experiment-hero

E091: Partial Euler products on the critical line
```

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

- Compare \(\zeta(1/2+it)\) to partial Euler products
  \(\prod_{p\le P} (1-p^{-s})^{-1}\)
  at a fixed `t` while increasing the prime cutoff `P`.
- Plot the mismatch to illustrate non-convergence on the critical line.

## Notes

- The Euler product is only convergent for \(\Re(s)>1\); this experiment visualizes the failure mode at \(\Re(s)=1/2\).

## Published run snapshot

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

```{include} ../reports/e091.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e091.json
:language: json
```

:::

## References

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

## Related experiments

- {doc}`e114` (E114: ζ via η: stability map on the critical line)
- {doc}`e092` (E092: 1/ζ(s) via the Möbius Dirichlet series)
- {doc}`e093` (E093: −ζ′(s)/ζ(s) via the von Mangoldt series)
- {doc}`e083` (E083: Series vs. Euler product (ζ))
- {doc}`e110` (E110: Dirichlet L-series partial sums at s=1 and s=1/2)