E090: Functional equation residual heatmap

E090 hero

E090: Functional equation residual heatmap

Tags: analysis, quantitative-exploration, visualization, riemann-zeta, numerics

Highlights

  • Focused numeric experiment with a single main figure.

  • Parameters saved to params.json for reproducibility.

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

What is computed

  • Evaluate the functional-equation residual (R(s)=\zeta(s)-\chi(s),\zeta(1-s)) on a small rectangle on the complex plane.

  • Visualize log10(|R(s)| + eps) as a heatmap.

Notes

  • This is a sanity-check style plot: away from poles/zeros and with sufficient precision, the residual should be small.

  • Parameters control precision and grid size.

Published run snapshot

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

We visualize log10|zeta(s) - chi(s) zeta(1-s)| on a coarse grid. Away from the pole at s=1, the residual should be close to numerical precision.

params.json (snapshot)
{
  "mp_dps": 70,
  "n_sigma": 25,
  "n_t": 31,
  "sigma_max": 2.0,
  "sigma_min": -1.0,
  "t_max": 15.0,
  "t_min": -15.0
}

References

  • See the zeta / Dirichlet-series references in refs.bib.