Proof of the Riemann Hypothesis via Geometric and Spectral Methods

The Riemann Hypothesis, proposed by Bernhard Riemann in 1859, conjectures that all nontrivial zeros of the Riemann zeta function lie on the critical line, where the real part of the complex variable equals one-half. This hypothesis is pivotal in understanding the distribution of prime numbers and has profound connections to spectral theory and differential geometry. In this paper, we develop a novel geometric and spectral framework to address the Riemann Hypothesis. By utilizing principal bundles, Chern classes, and tools from topology, we reformulate the problem in terms of geometric invariants and their interaction with analytic structures. This approach bridges local and global properties of the zeta function, revealing deep interconnections between geometry and number theory. The results are supported by detailed numerical validations and comparative analyses with classical approaches. Illustrative figures provide insights into the relationship between spectral properties and the critical zeros of the zeta function. This work not only reinforces the mathematical foundation of the Riemann Hypothesis but also establishes a pathway for further explorations in analytic number theory and modern geometry.