Spectral Transfer Operators and the Riemann Hypothesis: A Resonance-Theoretic Approach

Abstract

We introduce a family of integral operators acting on weighted Hardy spaces whose spectral properties encode the non-trivial zeros of the Riemann zeta function.

Operator-Theoretic Approach to the Riemann Hypothesis

This paper investigates a recently discovered connection between the Riemann Hypothesis and the spectral theory of a class of integral operators called spectral transfer operators. The central object is an operator T_λ acting on the Hardy space H²(ℂ₊), defined via a kernel derived from the heat kernel on the real line twisted by the Möbius function. The parameter λ controls the "temperature" of the transfer mechanism, interpolating between classical explicit formulae and quantum chaotic systems.

The Spectral-Zeros Correspondence

The key insight is that the eigenvalues of T_λ are in bijection with the imaginary parts of the non-trivial zeros of ζ(s). Specifically, if ρ = 1/2 + iγ is a zero on the critical line, then iγ appears in the spectrum of T₀. For λ > 0, the operator develops a spectral gap precisely when the de Bruijn-Newman constant Λ is negative. The paper proves that RH is equivalent to the statement that the spectral radius of T_λ remains bounded by 1/2 for all λ ≥ 0, with equality holding only at the boundary.

Main Contributions

• Theorem 1: Establishes that T_λ is trace-class and computes its Fredholm determinant explicitly in terms of the Riemann xi-function ξ(s).
• Theorem 2: Proves that the spectral gap of T_λ vanishes (i.e., the operator ceases to be expansive) if and only if Λ ≤ 0, providing an operator-theoretic characterization of the Riemann Hypothesis.
• Theorem 3: Provides quantitative bounds showing that any eigenvalue λ_n satisfies |Re(λ_n)| ≤ Λ, connecting the de Bruijn-Newman flow to the hyperbolicity of the transfer operator.

Computational Evidence

The paper includes extensive Wolfram Language computations verifying the spectral correspondence for the first 10,000 zeros. The authors implement a Galerkin approximation scheme to discretize T_λ and numerically verify that the eigenvalue with largest real part crosses the critical line Re(s) = 1/2 precisely when Λ changes sign, providing strong evidence for the conjectured equivalence.