E110: Dirichlet L-series partial sums at s=1 and s=1/2

E110 hero

E110: Dirichlet L-series partial sums at s=1 and s=1/2

Tags: number-theory, l-functions, quantitative-exploration, 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 partial Dirichlet series behavior near s=1 and on the critical line.

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_L_partial_magnitude.png

  • params.json

  • report.md

Notes

  • Using first nonprincipal character mod q.

  • Final partial sum at N=n_max: (0.8646862659836099+0.20411306614158545j).

  • For nonprincipal χ and Re(s)=1, the series converges (slowly).

params.json (snapshot)
{
  "n_max": 5000,
  "q": 5,
  "s_re": 1.0
}

References

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