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Spectral Geometry of Arithmetic Moduli Stacks and the Hilbert-Pólya Program

This paper investigates the spectral theory of canonical Laplacians acting on $L^2$-spaces of arithmetic moduli stacks of framed vector bundles over $\text{Spec}(\mathbb{Z})$, based on the framework of arXiv:mathematics_2601_13726v1.

Abstract

This paper investigates the spectral theory of canonical Laplacians acting on $L^2$-spaces of arithmetic moduli stacks of framed vector bundles over $\text{Spec}(\mathbb{Z})$, based on the framework of arXiv:mathematics_2601_13726v1.

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

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