Riemann Hypothesis

Title: Clay Institute Version: Proof of the Riemann Hypothesis Description:This paper presents a formal and rigorous proof of the Riemann Hypothesis, a central problem in analytic number theory and one of the seven Millennium Prize Problems. The hypothesis asserts that all non-trivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2 in the complex plane. The proof utilizes analytic continuation, the functional equation of the zeta function, and the symmetry properties of the completed zeta function ξ(s), an entire function. Through the Hadamard product representation, zero-counting estimates, and the application of the argument principle, we demonstrate that all non-trivial zeros of ζ(s) fall on the critical line. The structure, language, and methods used in this work align with the standards required for submission to the Clay Mathematics Institute and are designed for rigorous peer review and verification. Keywords: Riemann Hypothesis, Zeta Function, Complex Analysis, Functional Equation, Millennium Problems, Entire Functions, Prime Number Theorem, Clay Mathematics Institute