E069: L(1,χ): slow convergence and smoothing.

Preview figure for E069
Preview figure for E069

Tags: number-theory, quantitative-exploration, visualization, dirichlet-characters, l-functions

Highlights

  • Computes truncated Dirichlet L-series for a nontrivial character.

  • Compares series truncation against Euler-product truncation (where meaningful).

  • Shows convergence behavior (fast for s>1, slow at s=1).

What this experiment does

At s=1, the Dirichlet series for L(1,χ) converges very slowly. For the nontrivial character modulo 4:

The implementation focuses on a compact, reproducible numerical workflow: deterministic parameter defaults, structured output folders, and one or more figures saved for the gallery.

Outputs

This experiment writes into out/e069/:

  • figures/fig_01_l1_partial_sums.png

  • figures/fig_02_l1_smoothed.png

How to run

make run EXP=e069

Notes

  • The gallery preview figure shipped with the documentation uses conservative cutoffs so builds stay fast. If you run the experiment locally, increase the cutoffs to see the asymptotic regime more clearly.

  • Prime-race plots depend on the chosen sampling of x (linear vs. log grid). The qualitative “who leads” picture can change when you zoom in.

Published run snapshot

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

  • q: 4

  • n_max: 300000

  • last raw partial sum error vs pi/4: 1.667e-06

  • smoothed scales: [2000, 8000, 32000]

Figures:

  • fig_01_l1_partial_sums.png

  • fig_02_l1_smoothed.png

params.json (snapshot)
{
  "n_max": 300000,
  "q": 4,
  "smooth_scales": [
    2000,
    8000,
    32000
  ]
}

References

Apostol [1976], Montgomery and Vaughan [2006]