E080: Chebyshev bias: leader fraction vs. x.¶
Tags: number-theory, quantitative-exploration, visualization, prime-races
Highlights¶
Estimates the fraction of x where one residue class leads another.
Plots a running bias estimate over log-sampled x values.
Detects coarse sign-change neighborhoods for the race difference.
What this experiment does¶
For the mod 4 race D(x)=π(x;4,3)-π(x;4,1), define the empirical leader fraction:
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/e080/:
figures/fig_01_bias_fraction.png
How to run¶
make run EXP=e080
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).
x_max: 12000000
n_points: 4000
final fraction (D>0): 0.941
Figure:
fig_01_bias_fraction.png
Notes:
This is sample-grid dependent; it is a qualitative bias indicator, not a rigorous density.
params.json (snapshot)
{
"n_points": 4000,
"x_max": 12000000
}