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

E093 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).

We approximate -zeta’(s)/zeta(s) by partial sums of the von Mangoldt Dirichlet series.

params.json (snapshot)
{
  "mp_dps": 80,
  "n_values": [
    10,
    30,
    100,
    300,
    1000,
    3000,
    10000,
    30000
  ],
  "s": 2.0
}

References

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