E083: Series vs. Euler product (ζ)

E083 hero

E083: Series vs. Euler product (ζ)

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_vs_euler_product.png — main figure

Published run snapshot

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

We compare partial approximations to zeta(s) coming from the series and the Euler product.

Note: both approximations converge for Re(s) > 1, but their practical behavior depends on the cutoff choices.

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

References

Edwards [1974], Titchmarsh [1986]