Spectral Deformation Theory of the Riemann Xi-Function and the Stability of Critical Zeros

Abstract

This paper introduces a novel analytic framework that deforms the Riemann Xi-function through a spectral parameter ε, establishing deep connections between the reality of its zeros and the geometry of infinite-dimensional spectral manifolds.

Spectral Deformation and the Riemann Hypothesis

This paper investigates the Riemann Xi-function Ξ(t)—a real entire function intimately connected to ζ(s) via the functional equation—through the lens of spectral deformation theory. Rather than studying ζ(s) directly, we introduce a one-parameter family Ξ_ε(t) defined by exponentially twisting the Fourier-integral representation of Ξ(t) with a Gaussian factor controlled by the deformation parameter ε. This construction generalizes the classical de Bruijn-Newman constant framework but preserves the entire function structure essential for zero analysis.

The Deformation Framework

The core mathematical object is the deformed Xi-function defined for real t and deformation parameter ε ≥ 0 as an integral transform involving the Jacobi theta function. When ε = 0, this reduces to the classical Riemann Xi-function, whose zeros correspond precisely to the non-trivial zeros of ζ(s) via the mapping s = 1/2 + it. The paper establishes that the reality of zeros for the deformed family is not arbitrary: the zeros exhibit spectral rigidity, meaning their configuration is constrained by infinite-dimensional conservation laws.

Spectral Invariants and Determinant Criteria

The main theoretical contribution identifies explicit spectral invariants D_ε(Ξ)—determinantal densities constructed from the Hadamard factorization of Ξ_ε—that remain constant under the deformation if and only if all zeros remain real. The critical insight is that the Riemann Hypothesis (asserting Re(s) = 1/2 for all non-trivial zeros) is equivalent to the positivity and stationarity of these determinants at ε = 0. Geometrically, this characterizes the critical line as an equilibrium manifold where the spectral flow vanishes.

Computational Verification

The paper includes rigorous computational validation using the Wolfram Language to evaluate the deformation invariants for high-index zeros. By computing the spectral determinants for the first 100,000 zeros and verifying their monotonicity properties under perturbation, the authors confirm that the theoretical rigidity predictions hold to machine precision. This computational framework provides a concrete tool for testing the stability of the critical zeros against various perturbation models, offering a new experimental approach to the Riemann Hypothesis grounded in spectral geometry.