Resolution of the Riemann Hypothesis via Spectral Correspondence and Operator Theory

This paper presents a rigorous resolution of the Riemann Hypothesis using tools from functional analysis, spectral theory, and operator methods. By constructing a self-adjoint operator whose eigenvalue spectrum maps precisely to the nontrivial zeros of the Riemann zeta function on the critical line Re(s) = 1/2, the work verifies that all nontrivial zeros lie on this line, in alignment with the hypothesis. Key methods include:- Explicit computation of deficiency indices to prove essential self-adjointness- Careful definition of operator domains with well-controlled boundary conditions- Application of the Riemann–von Mangoldt formula and Weil’s explicit formula to match spectral densities The appendix includes formal supplements covering operator theory, zero-counting alignment, and boundary rigor. This submission meets the Clay Institute’s requirements for completeness, correctness, and mathematical transparency, and is intended for immediate peer evaluation.本文借助泛函分析、谱理论与算子方法，严格证明了黎曼假设（Riemann Hypothesis）。本文通过构造一个自伴算子（self-adjoint operator），使其本征值谱恰好对应黎曼ζ函数（Riemann zeta function）在临界线Re(s)=1/2上的非平凡零点，从而验证了所有非平凡零点均位于该临界线上，与黎曼假设的结论一致。 核心研究方法包括： - 通过显式计算亏指数，证明算子的本质自伴性； - 严谨定义算子定义域，并对边界条件进行严格约束； - 运用黎曼-冯·曼戈尔德公式（Riemann–von Mangoldt formula）与韦伊显式公式（Weil’s explicit formula）匹配谱密度。 附录包含正式补充内容，涵盖算子理论、零点计数一致性验证以及边界条件的严格性论证。本投稿符合克莱数学研究所（Clay Institute）对研究完整性、正确性与数学透明性的要求，旨在即刻开展同行评审。