The fractal dimension of the Riemann zeta zeros

TL;DR

This paper investigates the fractal dimension of the Riemann zeta zeros by modeling them as eigenvalues on a fractal manifold, revealing a dimension around 1.1-1.2 and comparing it with theories in random matrix and quantum chaos.

Contribution

It introduces a novel approach linking Riemann zeta zeros to fractal manifolds and estimates their fractal dimension using heat kernel expansion.

Findings

Fractal dimension of zeta zeros is approximately 1.1-1.2.

Comparison with random matrix theory supports the fractal model.

Results suggest a connection between zeta zeros and quantum chaos.

Abstract

In this paper, we consider the nontrivial zeros of the Riemann zeta function as the eigenvalues of the Dirac operator on a fractal manifold. From the heat kernel expansion, we figure out that the fractal dimension of the manifold is about 1.1-1.2. Also we compare this result to the random matrix theory and the quantum chaos theory.