E066: Character partial sums: cancellation profiles.¶
Tags: number-theory, quantitative-exploration, visualization, dirichlet-characters
Highlights¶
Constructs full character tables for a small modulus q.
Visualizes values as phases / real parts to spot structure and symmetries.
Checks orthogonality and basic sanity constraints numerically.
What this experiment does¶
For a Dirichlet character χ modulo q, consider partial sums:
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/e066/:
figures/fig_01_max_partial_sums.pngfigures/fig_02_example_partial_sum.png
How to run¶
make run EXP=e066
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).
Parameters¶
q: 15
N_max: 50000
include_principal: False
Summary¶
Computed partial sums S(N)=sum_{n<=N} chi(n) for 7 Dirichlet characters modulo q.
Top characters by max |S(N)|¶
| rank | index | max |S(N)| | conductor | principal | |—:|—:|—:|—:|:—:| | 1 | 5 | 3 | 15 | False | | 2 | 0 | 2 | 5 | False | | 3 | 3 | 2 | 3 | False | | 4 | 2 | 2 | 5 | False | | 5 | 6 | 1.41421 | 15 | False | | 6 | 4 | 1.41421 | 15 | False | | 7 | 1 | 1 | 5 | False |
params.json (snapshot)
{
"include_principal": false,
"n_max": 50000,
"q": 15,
"top_k": 8
}