# E092: 1/ζ(s) via the Möbius Dirichlet series

```{figure} ../_static/experiments/e092_hero.png
:alt: E092 hero
:class: experiment-hero

E092: 1/ζ(s) via the Möbius Dirichlet series
```

**Tags:** `analysis`, `quantitative-exploration`, `visualization`, `riemann-zeta`, `mobius`, `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 \(1/\zeta(s)\) to the partial sums
  \(\sum_{n\le N} \mu(n)/n^s\).
- Plot convergence as `N` increases, and highlight dependence on `Re(s)`.

## Notes

- This is the classic identity \(\sum_{n\ge1} \mu(n)/n^s = 1/\zeta(s)\) (absolute convergence for \(\Re(s)>1\)).

## Published run snapshot

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

```{include} ../reports/e092.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e092.json
:language: json
```

:::

## References

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

## Related experiments

- {doc}`e093` (E093: −ζ′(s)/ζ(s) via the von Mangoldt series)
- {doc}`e110` (E110: Dirichlet L-series partial sums at s=1 and s=1/2)
- {doc}`e111` (E111: Euler product vs. Dirichlet series for L(s,χ))
- {doc}`e091` (E091: Partial Euler products on the critical line)
- {doc}`e082` (E082: Zeta(s) series convergence)