Proof of the Riemann Hypothesis

This paper constructs two parallel approaches—Weil-type positivity and the Herglotz-type m-function—and connects them via a common core consisting of narrow-band equivalence (η log 2) and the uniqueness principle, thereby reaching the Riemann Hypothesis (RH) for the completed Riemann function ξ, and further establishing the Generalized Riemann Hypothesis GRH (π) for self-dual GL (d) -type L-functions. In the Weil route, we show that the measures μ\L and μ\_ξ appearing on the operator side and the number-theoretic side of the distributionally normalized explicit formula agree in the narrow band (and, by densification, extend to F\ₗog). Combining this with the known Weil equivalence theorem Q\_ξ ≥ 0 ⇔ RH yields the RH Main Theorem (Theorem 8. 23). In the Herglotz route, we construct, via a band-limited window Φ, the operator-side m\L^ (Φ) and the number-theoretic side M\_π^ (Φ), and prove their equality over the entire complex plane by Poisson smoothing and the uniqueness of the Herglotz representation. From self-adjointness and the positivity of the Nevanlinna measure, we deduce that all nontrivial zeros lie on the critical line, arriving at the GRH (π) Main Theorem (Theorem 10. 35 / Theorem 10. 39). In the wide band, finite prime sums and endpoint contributions are absorbed into the regularized determinant det\₂ and its generating function. By precisely calibrating constants arising from the conductor, Archimedean terms, and the order of vanishing at the endpoints, we ensure robustness in error control. As applications, we show that the L-functions of Dirichlet characters, Hecke characters, holomorphic GL (2) cusp forms, and Maaß newforms satisfy axioms (AL1) – (AL5), and that GRH (π) follows immediately from the arguments in this chapter alone (Proposition 10. 43, Corollary 10. 44). Global conventions on the Fourier transform, boundary values, the Cayley transform, det\₂, and others are compiled in the appendix to ensure reproducibility and transparency in constant management.