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

E092 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).

We approximate 1/zeta(s) via the Dirichlet series sum mu(n)/n^s, which converges for Re(s) > 1.

params.json (snapshot)
{
  "mp_dps": 70,
  "n_values": [
    10,
    30,
    100,
    300,
    1000,
    3000,
    10000,
    30000
  ],
  "s": 2.0
}

References

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