E118: Chebyshev bias: lead-time statistics

E118 hero

E118: Chebyshev bias: lead-time statistics

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

Highlights

  • Focused numeric experiment with a small set of figures.

  • Parameters saved to params.json for reproducibility.

  • Defaults are chosen, so the experiment remains feasible for the CI “slow” suite.

What is computed

  • Measure how often one residue class leads another and visualize lead-time statistics.

Notes

  • This page summarizes the intent; see the generated report.md in out/ for concrete outputs.

Published run snapshot

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

Artifacts

  • figures/fig_01_euler_product_breakdown.png

  • params.json

  • report.md

Notes

  • Best (min) abs error per s: {‘(1.2+0j)’: ‘2.14e-01’, ‘(1.05+0j)’: ‘8.26e+00’, ‘(0.9+0j)’: ‘1.83e+01’}

  • As Re(s) approaches 1 from the right, convergence slows; for Re(s)<=1 the Euler product no longer converges.

params.json (snapshot)
{
  "dps": 60,
  "p_max": 50000,
  "p_steps": 40,
  "s_values": [
    {
      "imag": 0.0,
      "real": 1.2
    },
    {
      "imag": 0.0,
      "real": 1.05
    },
    {
      "imag": 0.0,
      "real": 0.9
    }
  ]
}

References

  • See docs/background/primes-in-arithmetic-progressions.md and docs/background/prime-number-races.md.