Gram points and zero counting¶

A Gram point is (informally) a value (t) where the Riemann–Siegel theta function (\theta(t)) hits an integer multiple of (\pi). These points are useful landmarks when visualizing (Z(t)) and tracking sign changes.

The Riemann–von Mangoldt formula gives the asymptotic count of nontrivial zeros up to height (T):

\[ N(T)=\frac{T}{2\pi}\log\!\left(\frac{T}{2\pi}\right)-\frac{T}{2\pi}+O(\log T)\quad (T\to\infty), \]

with refinements that include the argument of ζ on the critical line.

Key ideas¶

Bookkeeping: zero counting provides a “sanity check” for numerical root-finding: we can compare a computed zero list against (N(T)).
Gram blocks / Gram’s law: empirical rules about how zeros sit between consecutive Gram points (useful, but not always true).
Practical numerics: many experiments become clearer when plotted against (\theta(t)) or indexed by Gram points.

Experiments in this repository¶

E116 — Observed zero counts (via sign changes) vs Riemann–von Mangoldt estimate.

References¶

See References.

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