Probabilistic models for Gram's Law

TL;DR

This paper introduces a probabilistic model to estimate the frequency of Gram's Law pattern occurrences in the zeros of the Riemann zeta function, linking zero distribution to random matrix eigenvalues.

Contribution

It proposes a novel model connecting zeta zeros to random matrix theory to predict Gram's Law occurrences.

Findings

Model estimates the frequency of Gram's Law validity.

Links zeta zero distribution to eigenvalues of random unitary matrices.

Provides a probabilistic framework for understanding zero patterns.

Abstract

Gram's Law describes a pattern that frequently occurs in the distribution of the non-trivial zeros of the Riemann zeta function along the critical line. Whenever Gram's Law holds true, it reduces the difficulty of computing the corresponding zeta zeros. In this paper, we provide a model that estimates how often this pattern occurs. The model is based on a conjecture that relates the statistical distribution of the zeta zeros to that of the eigenvalues of random unitary matrices.