E082: Zeta(s) series convergence

E082 hero

E082: Zeta(s) series convergence

Tags: analysis, quantitative-exploration, visualization, riemann-zeta, numerics

Highlights

  • Focused numeric experiment with a single main figure.

  • Parameters saved to params.json for reproducibility.

  • Lightweight computation suitable for CI “slow” suite (small defaults).

What is computed

  • A parameterized numeric evaluation related to the Riemann zeta function.

  • A visualization summarizing the main phenomenon for the chosen parameter range.

Algorithm sketch

  1. Build the numeric grid / sampling points.

  2. Evaluate the target quantity with controlled truncation.

  3. Render the figure and write a short report.

Outputs

  • report.md — short narrative summary

  • params.json — experiment parameters snapshot

  • figures/fig_01_series_convergence.png — main figure

Published run snapshot

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

We approximate zeta(s) by partial sums of the Dirichlet series for two real s values.

  • As s approaches 1, convergence becomes much slower.

  • The plot shows the absolute error relative to a high-precision mpmath zeta(s).

params.json (snapshot)
{
  "mp_dps": 60,
  "n_values": [
    10,
    30,
    100,
    300,
    1000,
    3000,
    10000,
    30000
  ],
  "s_values": [
    2.0,
    1.2
  ]
}

References

Edwards [1974], Titchmarsh [1986]