We construct a family of self-adjoint operators HN on Hilbert spaces associated with quantum graphs whose vertices are placed at the logarithms of the first N primes, xₚ = ln p. The matching conditions at the vertices are unitary and encode the local scaling symmetry of the p-adic fields Qₚ. We prove that the spectral determinant of HN satisfies det (HN - λI) ∝ ξN (s), where s = 1/2 + iλ and ξN (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.
Sleiman Nisrin (2026) studied this question.