E068: Dirichlet L(s,χ): series vs. Euler product (partial approximations).¶
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¶
For Re(s) > 1, Dirichlet L-functions admit both:
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/e068/:
figures/fig_01_series_vs_euler.png
How to run¶
make run EXP=e068
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
s: (2+0j)
n_max: 120000
p_max: 400000
|series - euler| (last sample): 1.170e-10
Figure:
fig_01_series_vs_euler.png
Notes:
Both approximations converge to the same limit for Re(s)>1.
The Euler product is only over primes p not dividing q.
params.json (snapshot)
{
"n_max": 120000,
"n_points": 160,
"p_max": 400000,
"q": 4,
"s_im": 0.0,
"s_re": 2.0
}