E086: Hardy Z(t) near zeros

E086 hero

E086: Hardy Z(t) near zeros

Tags: analysis, quantitative-exploration, visualization, riemann-zeta, zeta-zeros, critical-line, numerics

Highlights

  • Focused numeric experiment with a single main figure.

  • Parameters saved to params.json for reproducibility.

  • Lightweight computation suitable for CI “slow” suite (small defaults).

What is computed

  • A parameterized numeric evaluation related to the Riemann zeta function.

  • A visualization summarizing the main phenomenon for the chosen parameter range.

Algorithm sketch

  1. Build the numeric grid / sampling points.

  2. Evaluate the target quantity with controlled truncation.

  3. Render the figure and write a short report.

Outputs

  • report.md — short narrative summary

  • params.json — experiment parameters snapshot

  • figures/fig_01_hardy_Z_near_zeros.png — main figure

Published run snapshot

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

We locate the first few imaginary parts of non-trivial zeros (via mpmath) and sample Hardy’s Z(t) in a small window around each zero.

params.json (snapshot)
{
  "mp_dps": 60,
  "n_points": 400,
  "n_zeros": 3,
  "window": 0.6
}

References

Edwards [1974], Titchmarsh [1986]