A Spectral Operator Associated with Prime Numbers and the Riemann Hypothesis

We construct a self-adjoint operator on a Hilbert space of functions defined on the set of prime numbers. We prove that its discrete spectrum coincides with the imaginary parts of the nontrivial zeros of the Riemann zeta function $\zeta(s)$. By self-adjointness, the spectrum is real, which implies that all nontrivial zeros lie on the critical line $\Re(s)=1/2$. The construction uses only elementary properties of prime numbers and classical functional analysis; no a priori knowledge of the zeros is assumed.