On the Generalized Riemann Hypothesis for Selberg Zeta Functions via Isometric Projections on Hyperbolic Surfaces

Generalized is the method of self-adjoint projection operators from Euclidean domains to theLaplace-Beltrami framework on a compact Riemann surface X=Γ /H 2 of genus g ≥2.Constructed is an infinite-dimensional isometric projection operator P_ Γ that systematicallyannihilates the modular fluctuations induced by the length spectrum of closed geodesics on ( X ) .Proved is, that the compressed hyperbolic Hamiltonian H_ Γ =P_ Γ Δ_ X P _Γ is densely defined andstrictly self-adjoint. Consequently, its discrete spectrum is rigidly bound to the real axis,establishing that all non-trivial zeroes of the Selberg zeta function Z ( s ) lie precisely on thecritical line R ( s )=1/2, thereby proving the Generalized Riemann Hypothesis for arithmeticFuchsian groups