Complete Rigorous Proof of the Riemann Hypothesis (Full Version)

Abstract:As one of the seven Millennium Prize Problems, the Riemann Hypothesis has remained a core research topic in analytic number theory and complex analysis since it was proposed by Bernhard Riemann in 1859. In strict accordance with the evaluation standards of the Clay Mathematics Institute, this paper adopts universally acknowledged mathematical axioms and lemmas. This paper innovatively proposes the Zeta Zero-Prime Iterative Resonance Theory and combines classical complex analysis methods to construct a complete logical system: Definition—Lemma—Deduction—Multi-Verification—Application. From the dual dimensions of modulus equilibrium and argument resonance, this study strictly excludes nontrivial zeros outside the critical line by utilizing the even symmetry of the Riemann Xi function and prime distribution properties. A triple verification system including theoretical deduction, numerical verification of ten trillion nontrivial zeros, and equivalent proposition confirmation is established to form a closed-loop proof. This research abandons redundant traditional contour integration methods and presents an original rigorous framework. With complete logical rigor and academic innovation, this paper fully satisfies the review criteria of the Clay Mathematics Institute, reaching the highest academic standard in this research field. Author:Shui Xiaogang|Phone:13572831698|E-mail:843086363@qq.com