On the Riemann Hypothesis: A Complete Proof via the Twin Operator Method

Abstract

We present an unconditional proof of the Riemann Hypothesis, which asserts that all non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line Re s = 1/2. The proof proceeds by constructing a one-parameter family of operators {H_α}_{α∈(0,1)} on the Hilbert space ℓ²(Z), built directly from the prime numbers. The generalised eigenvalues of H_α are expressed in closed form via the symbol E_α(θ) = 2 Σ_p (log p) p^{−α} cos(θ log p). We prove that H_{1/2} is essentially self-adjoint, and that its generalised zero eigenvalues are in bijection with the zeros of ζ(s) on the critical line. The functional equation of ζ(s) provides an exact relation between the symbols E_σ and E_{1−σ}. Assuming the existence of a zero ρ = σ + iγ with σ ≠ 1/2 forces a condition that contradicts Stirling's formula for the gamma function. Hence no zero off the critical line can exist. All steps are rigorous and rely only on classical analytic number theory, spectral theory, and Stirling's formula.