# E093: −ζ′(s)/ζ(s) via the von Mangoldt series

```{figure} ../_static/experiments/e093_hero.png
:alt: E093 hero
:class: experiment-hero

E093: −ζ′(s)/ζ(s) via the von Mangoldt series
```

**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

- For \(\Re(s)>1\), compare \(-\zeta'(s)/\zeta(s)\) to the partial sums
  \(\sum_{n\le N} \Lambda(n)/n^s\).
- Plot the truncation error as a function of `N`.

## Notes

- Uses the von Mangoldt function \(\Lambda(n)\); the series converges absolutely for \(\Re(s)>1\).

## Published run snapshot

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

```{include} ../reports/e093.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e093.json
:language: json
```

:::

## References

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

## Related experiments

- {doc}`e092` (E092: 1/ζ(s) via the Möbius Dirichlet series)
- {doc}`e091` (E091: Partial Euler products on the critical line)
- {doc}`e082` (E082: Zeta(s) series convergence)
- {doc}`e083` (E083: Series vs. Euler product (ζ))
- {doc}`e088` (E088: Zero counting via Riemann–von Mangoldt)