通过谱几何方法证明黎曼猜想

Abstract On the hyperbolic surface Y = Γ\ℍ, we introduce a Kato type-(A) holomorphic family of operators on the complex Hilbert space 𝓗 ≃ L²(Y, S), namely K(z) = iΓD + iz, where z ∈ ℂ. Using techniques from pseudodifferential operators and ζ-regularized determinants, we first obtain the reflection identity det_{ζ,α}[−K(z̅)] = (det_{ζ,α} K(z))̅ and the spectral parameters z = ±i|λ_k|. By analyzing the relationship between the symmetry transformation in this operator reflection identity and the symmetry in the functional equation of the Riemann ζ(s) function, we establish the biholomorphic map z = Φ(s) = ±iR exp(i(s − 1/2)), where s ∈ ℂ and R ∈ ℝ_{>0}. With the aid of the Selberg trace formula and related theory, we construct a correspondence between the spectrum of det_{ζ,α} K(z) and the nontrivial zeros of the Riemann ζ(s) function. Finally, combining this with the fact that the nontrivial zeros lie in the critical strip {ρ ∈ ℂ | 0 < Re ρ < 1}, we rigorously and completely prove that all nontrivial zeros of the Riemann ζ(s) function lie on the critical line Re ρ = 1/2.