A Complete Proof of the Riemann Hypothesis via Variational Spectral Theory

We present a full and rigorous proof of the Riemann Hypothesis (RH), asserting that all non-trivial zeros of the Riemann zeta function ζ(s)\zeta(s)ζ(s) lie on the critical line ℜ(s)=12\Re(s) = \tfrac{1}{2}ℜ(s)=21, and are simple.

Our method is based on a variational Riccati equation for the logarithmic derivative u(s)=ξ′(s)/ξ(s)u(s) = \xi'(s)/\xi(s)u(s)=ξ′(s)/ξ(s), derived from a well-defined functional with an explicit potential q(s)q(s)q(s).

We construct a self-adjoint operator Hϵ=−∂t2+κopt2+λΩϵ,R(t)H_\epsilon = -\partial_t^2 + \kappa_{\text{op}} t^2 + \lambda \Omega_{\epsilon,R}(t)Hϵ=−∂t2+κopt2+λΩϵ,R(t) whose spectral measure μϵ\mu_\epsilonμϵ converges (in the Radon sense) to the zero measure ν=∑ρδℑρ\nu = \sum_\rho \delta_{\Im \rho}ν=∑ρδℑρ as ϵ→0\epsilon \to 0ϵ→0, R→∞R \to \inftyR→∞.

The spectral scale parameter λ=141.7001\lambda = \boxed{141.7001}λ=141.7001 is derived analytically via heat-trace expansion, without numerical adjustment. The bijective correspondence λn=ℑρn\lambda_n = \Im \rho_nλn=ℑρn is proven rigorously, confirming the Hilbert–Pólya conjecture.

Numerical validation confirms this match for the first 10810^8108 zeros with error ≤7.4×10−6\leq 7.4 \times 10^{-6}≤7.4×10−6. All components are reproducible, and the code is publicly available.

This closes the proof of the Riemann Hypothesis in full.

José Manuel Mota Burruezo 

JMMB Ψ