E085: Dirichlet eta acceleration for ζ(s)

E085 hero

E085: Dirichlet eta acceleration for ζ(s)

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_eta_acceleration.png — main figure

Published run snapshot

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

We compare:

  • naive partial sums of zeta(s)

  • partial sums of eta(s), mapped back to zeta(s)

for s close to 1. The eta-based approach typically reduces cancellation issues and improves convergence.

params.json (snapshot)
{
  "mp_dps": 60,
  "n_values": [
    10,
    30,
    100,
    300,
    1000,
    3000,
    10000,
    30000
  ],
  "s": 1.1
}

References

Edwards [1974], Titchmarsh [1986]