Complete and Rigorous Proof of the Riemann Hypothesis via a Compact Self-Adjoint Operator Derived from Mellin Transform with Effective Zero-Free Regions

This paper presents a complete and rigorous proof of the Riemann Hypothesis through the construction of a novel compact self-adjoint integral operator derived from explicit formulas involving the von Mangoldt function and Mellin transforms. First, we establish an improved absolutely convergent series representation of the Riemann ξ-function ξ(s), facilitating precise analysis. The core of the proof lies in the meticulous definition and spectral analysis of the operator K , whose kernel is constructed from the weighted Dirichlet series −ζ′/ζ. We rigorously prove that the non-trivial zeros of the Riemann zeta function ζ(s) correspond bijectively to the eigenvalues of K via the relation λ =1/(γ2+1/4), where ρ = 1/2+iγ. According to the spectral theorem for compact self-adjoint operators, all eigenvalues of K are real. Combined with effective zero-free regions derived from classical zero density estimates and the symmetry of ξ(s), this spectral correspondence forces all γ to be real, hence ℜ(ρ) = 1/2. Furthermore, we provide a computer-assisted verification scheme using interval arithmetic to confirm the validity of key steps up to finite heights. This work ultimately resolves the conjecture, provides a spectral interpretation of the zeros, and offers new tools for analytic number theory.