Proof of the Riemann Hypothesis

Background: The non-trivial zeros of the Riemann zeta function govern prime distributions, and the Riemann Hypothesis states that they all lie on the critical line $\operatorname{Re}s=\tfrac12$. Methods: We place a self-adjoint restriction $R_{\mathrm{PW}}$ of the first-order differentiation operator on a weighted Hilbert space $H_\alpha$ and, under the Paley–Wiener band-limit $\Lambda=\pi$, obtain a Hilbert–Schmidt kernel $K$. Its discrete spectrum $(\gamma_k)$ defines candidate zeros $s_k=\tfrac12+i\gamma_k$. A Montgomery–Odlyzko gap bound combined with the Guinand–Weil explicit formula yields the counting identity $N_\zeta(T)=N_{\mathrm{eig}}(T)$. We further prove that the regularised Fredholm determinant $D(z)=\det_{2}(I+zK)$ satisfies $\xi(s)=D(i(s-\tfrac12))$ throughout the complex plane. Results: The injection together with exact counting shows that the eigenvalues and zeta zeros correspond bijectively; the reality of $\gamma_k$ forces $\operatorname{Re}s_k=\tfrac12$, thus establishing the Riemann Hypothesis. The determinant identity simultaneously ties the completed zeta function to operator theory. Conclusions: The paper provides a self-contained, purely analytic and operator-theoretic proof of the Riemann Hypothesis and outlines how the same framework can extend to the Selberg zeta and other $L$-functions.