E074: Prime race mod 8: leaderboard among 1,3,5,7.

Preview figure for E074
Preview figure for E074

Tags: number-theory, quantitative-exploration, visualization, prime-races, aps

Highlights

  • Computes prime-race differences π(x;q,a) − π(x;q,b) on a grid of x values.

  • Visualizes lead changes and the size of fluctuations as x grows.

  • Adds a derived statistic (normalization / θ-variant) to compare behaviors.

What this experiment does

This experiment tracks four residue classes modulo 8:

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/e074/:

  • figures/fig_01_leader_fractions.png

  • figures/fig_02_counts_minus_mean.png

How to run

make run EXP=e074

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: 3000000

  • residues: [1, 3, 5, 7]

  • n_points: 1000

  • log_grid: True

Leader fractions on the sample grid:

  • a=1: 0.029

  • a=3: 0.555

  • a=5: 0.262

  • a=7: 0.154

Figures:

  • fig_01_leader_fractions.png

  • fig_02_counts_minus_mean.png

Notes:

  • This uses a coarse sample grid, so fractions are approximate and depend on sampling.

params.json (snapshot)
{
  "log_grid": true,
  "n_points": 1000,
  "residues": [
    1,
    3,
    5,
    7
  ],
  "x_max": 3000000
}

References

Granville and Martin [2006], Rubinstein and Sarnak [1994]