E111: Euler product vs. Dirichlet series for L(s,χ)

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E111: Euler product vs. Dirichlet series for L(s,χ)

Tags: number-theory, l-functions, model-checking, visualization, dirichlet-series, numerics

Highlights

  • Focused numeric experiment with a small set of figures.

  • Parameters saved to params.json for reproducibility.

  • Defaults are chosen, so the experiment remains feasible for the CI “slow” suite.

What is computed

  • Compare truncated Euler products and truncated series for L(s,χ).

Notes

  • This page summarizes the intent; see the generated report.md in out/ for concrete outputs.

Published run snapshot

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

Artifacts

  • figures/fig_01_euler_product_error.png

  • params.json

  • report.md

Notes

  • Reference computed by series up to N=8000

  • Best (min) absolute error across P cutoffs: 4.123e-06

  • For Re(s)>1, Euler product and series converge to the same L(s,χ).

params.json (snapshot)
{
  "n_series": 8000,
  "p_max": 2000,
  "p_steps": 30,
  "q": 5,
  "s_re": 1.5
}

References

  • See docs/background/dirichlet-l-functions.md and docs/background/dirichlet-convolution.md.