We present a complete proof of the Riemann Hypothesis based on the fundamental structure of the Euler product and spectral theory of self-adjoint operators. Starting from the arithmetic fundamental theorem, we linearize the multiplicative structure of prime numbers and construct a sequence of finite-dimensional self-adjoint matrices that are strictly equivalent to the truncated Riemann xi function. Using mathematical induction, we prove that the eigenvalues of these matrices converge in order to the squares of the imaginary parts of the non-trivial zeros of the Riemann zeta function, and self-adjointness is strictly preserved under recursion. By the monotone convergence theorem for self-adjoint operators, we extend the finite-dimensional results to the infinite-dimensional case, proving the existence of a unique infinite-dimensional self-adjoint operator whose spectrum is exactly the set of squares of the imaginary parts of all Riemann zeros. Since the spectrum of a self-adjoint operator is necessarily real, we conclude that all non-trivial zeros of the Riemann zeta function have real part 1/2, establishing the Riemann Hypothesis.2020 Mathematics Subject Classification: 11M26, 47A10
Jianning Yang (Mon,) studied this question.