Hardy’s Z-function and the critical line

For real (t), the Hardy Z-function is defined (up to standard conventions) by

\[ Z(t)=e^{i\theta(t)}\,\zeta\!\left(\tfrac12+it\right), \]
where (\theta(t)) is the Riemann–Siegel theta function. The key feature is that Z(t) is real-valued for real t, so zeros of (Z(t)) correspond to zeros of ζ(s) on the critical line.

Key ideas

  • Real signal: studying sign changes of (Z(t)) is a practical way to bracket zeros on (\Re(s)=\tfrac12).

  • Theta function: (\theta(t)) captures the “oscillatory phase” of ζ on the critical line and is closely tied to Gram points.

  • Numerical workflows: many computational approaches to zeta zeros are formulated in terms of (Z(t)), (\theta(t)), and their approximations.

Why it matters in this project

It gives a clean, visualization-friendly path from “complex ζ-values” to “real curves” whose roots can be bracketed and counted.

Experiments in this repository

  • E115 — Hardy Z sign-change scan and zero bracketing (bisection refinement).

References

See References.

[Odlyzko, 1992, Titchmarsh, 1986, Wikipedia contributors, 2025, Wikipedia contributors, 2025]