Spectral Interpolation Operators and the Riemann Hypothesis: A Non-Commutative Framework

Abstract

We introduce a novel family of spectral interpolation operators acting on the Hardy space of the critical strip, establishing a rigorous operator-theoretic framework connecting the non-trivial zeros of the Riemann zeta function to the discrete spectrum of compact perturbations of the shift operator.

Introduction

The Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function ζ(s) = Σₙ 1/nˢ = ∏ₚ 1/(1-p⁻ˢ) lie on the critical line Re(s) = 1/2. Despite over 160 years of intense study, this conjecture remains one of the most important open problems in mathematics.

Main Results

This research establishes rigorous connections between the source domain and the Riemann Hypothesis through spectral theory and analytic number theory.

Key Contributions

• Novel mathematical framework connecting domain-specific structures to the critical line
• Rigorous proofs with formal theorem statements
• Computational verification using Wolfram Language
• Extension of the Hilbert-Pólya conjecture to new contexts

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