Spectral Determinants and the Riemann Hypothesis: Operator-Theoretic Bounds on the Critical Line

Abstract

We investigate a class of compact integral operators K_ρ acting on the Hardy space H²(C₊) whose Fredholm determinants encode the non-trivial zeros of the Riemann zeta-function.

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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