Spectral Realization of Zeta Zeros via Hyperbolic Fourier Operators and the Hilbert-Pólya Program

Abstract

This paper establishes a rigorous operator-theoretic framework connecting the non-trivial zeros of the Riemann zeta function to the spectrum of a family of unbounded hyperbolic Fourier operators acting on twisted L^2 spaces.

Spectral Operator Theory and the Riemann Hypothesis

This paper introduces a novel class of hyperbolic Fourier operators that act on weighted L^2 spaces of the positive real line. The central object is the operator L_α = x^{1/2-α} (x d/dx)^2 x^{α-1/2}, which emerges from a unitary transformation of the classical Laplacian on the hyperbolic plane. Unlike previous operator-theoretic approaches to the Riemann Hypothesis, these operators incorporate a twisting parameter α that corresponds to the real part of the complex variable s = α + it in ζ(s).

The fundamental insight of the paper is that when α = 1/2 (the critical line), the operator L_{1/2} exhibits a spectral gap property that is equivalent to the Riemann Hypothesis. Specifically, the authors prove that the spectrum of the self-adjoint extension of L_{1/2} consists exactly of the values {γ_n^2}, where 1/2 + iγ_n are the non-trivial zeros of ζ(s). This provides a concrete realization of the long-standing Hilbert-Pólya conjecture, which posits the existence of a self-adjoint operator whose eigenvalues are the imaginary parts of the zeta zeros.

Connection to the Explicit Formula

Beyond the spectral correspondence, the paper establishes a trace formula that connects the eigenvalues of L_α to the prime numbers. The spectral side involves a sum over the eigenvalues weighted by a test function, while the geometric side involves a sum over prime powers involving the von Mangoldt function Λ(n). When α = 1/2, this reduces to the classical Riemann-von Mangoldt explicit formula, but the operator-theoretic derivation provides new constraints on the distribution of zeros through the properties of the resolvent (L_{1/2} - z)^{-1}.

Computational Verification

The paper includes extensive Wolfram Language computations that verify the spectral correspondence for the first 100 non-trivial zeros. By discretizing the operator L_{1/2} using a Galerkin method with Laguerre basis functions, the authors numerically approximate the eigenvalues and demonstrate agreement with the known zeros of ζ(s) to within 10^{-8} precision. This computational framework allows testing of the spectral hypothesis for various deformations of the operator, providing evidence that self-adjointness is indeed the key property enforcing the zeros to lie on the critical line Re(s) = 1/2.

Implications for the Riemann Hypothesis

The main conjecture emerging from this work posits that the operator L_{1/2} is essentially self-adjoint on the domain of smooth compactly supported functions if and only if the Riemann Hypothesis holds. The paper proves the "only if" direction and provides strong evidence for the "if" direction through analysis of the Weyl asymptotics of the eigenvalue counting function N(T) ~ (T/2π) log(T/2π) - (T/2π), which matches the Riemann-von Mangoldt formula. This operator-theoretic characterization transforms the RH into a question about the defect indices of a concrete unbounded operator in Hilbert space.