Explicit Formulae and Spectral Asymptotics: A Geometric Criterion for the Riemann Hypothesis

Abstract

This paper establishes a novel equivalence between the Riemann Hypothesis and the optimal error bound in Weyl asymptotics for spectral counting functions of self-similar fractal strings.

Explicit Formulae and Fractal Spectra

This paper investigates a profound connection between the Riemann Hypothesis and the spectral theory of fractal strings. A fractal string is a one-dimensional vibrating object with fractal boundary, and its spectrum consists of the eigenvalues of the associated Laplacian operator. The authors focus specifically on self-similar fractal strings—geometric objects constructed via scaling ratios such as those derived from the golden ratio φ = (1+√5)/2.

The Spectral Explicit Formula

The central mathematical contribution is an explicit formula expressing the spectral counting function N(λ)—which counts eigenvalues not exceeding λ—as a sum of a dominant Weyl term (proportional to λ²) plus an infinite oscillatory sum over the non-trivial zeros ρ of the Riemann zeta function ζ(s). The formula takes the precise form N(λ) = (vol(ℒ)/2π)λ² + Σ_ρ (ζ_ℒ(ρ)/ρ)λ^ρ + O(λ^{1+ε}), where ζ_ℒ denotes the geometric zeta function of the fractal string. This reveals that the vertical distribution of zeta zeros directly controls the geometric oscillations in the spectral counting function.

A Geometric Criterion for the Riemann Hypothesis

The authors prove that the Riemann Hypothesis is equivalent to the bound R(λ) = O(λ^{1/4+ε}) on the spectral remainder term R(λ) = N(λ) − (Weyl term) for the specific golden ratio fractal string. Consequently, the hypothesis can be characterized as a statement about the optimal error term in Weyl asymptotics for this class of fractal geometries. If the Riemann Hypothesis were false, the explicit formula predicts that the remainder would exhibit oscillations of order λ^β where β equals the supremum of the real parts of zeta zeros, necessarily exceeding 1/2.

Computational Verification

The paper includes complete Wolfram Language code to compute the spectral counting function for the golden string and to verify the explicit formula numerically. By truncating the infinite sum over zeta zeros and comparing with direct eigenvalue counts, the authors provide computational evidence supporting the theoretical bounds. Numerical experiments confirm that the spectral remainder grows roughly like λ^{1/4}, consistent with the predictions of the Riemann Hypothesis and offering a novel algorithmic approach to studying the critical zeros through fractal spectral analysis.