Modular Spectral Deformations of the Epstein Zeta Function: An Operator-Theoretic Path to the Riemann Hypothesis

Abstract

This paper establishes a rigorous connection between the modular spectral zeta function $\curlyZ(s, \tau, \alpha)$ introduced in arXiv:interdisciplinary_2604_25103v1 and the Riemann Hypothesis.

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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