The Adelic Quantum Graph: A Self-Adjoint Realization of the Riemann Zeros

We construct a family of self-adjoint operators H_N on Hilbert spaces associated with quantum graphs whose vertices are placed at the logarithms of the first N primes, x_p = ln p. The matching conditions at the vertices are unitary and encode the local scaling symmetry of the p-adic fields Q_p. We prove that the spectral determinant of H_N satisfies det(H_N - λI) ∝ ξ_N(s), where s = 1/2 + iλ and ξ_N(s) is the truncated completed Riemann zeta function. Self-adjointness forces the eigenvalues λ (hence the zeros) to lie on the real line, i.e. Re(s)=1/2. As N→∞ the operators converge to a limit operator H_∞ whose pure point spectrum coincides with the non-trivial zeros of the Riemann zeta function.