Explicit Self-Adjoint Operator for the Riemann Zeros: Jacobi Matrix Construction, Cardano Formula for J3, and High-Precision Numerical Localisation

Abstract

We write down explicitly an infinite symmetric Jacobi matrix whose spectrum, under the Riemann Hypothesis, is the set {1/γ_n^2} where ρ_n=1/2+iγ_n are the non-trivial zeros of the Riemann zeta function. The construction proceeds through: (1) the three-term identity ζ(s)=Li_s(z_1)+Li_s(z_2)+∑ c_n/n^s at z_1=2−√2, z_2=√2−1; (2) Stieltjes moment problem on the discrete measure μ=∑ γ_n^{-2} δ_{γ_n^{-2}}; (3) Gram–Schmidt orthogonalisation yielding explicit Jacobi parameters α_n, b_n for n=0,…,13; (4) self-adjointness via Carleman condition; (5) Cardano closed form for the 3×3 truncation; (6) high-precision numerics at 120 digits recovering the first five zeros to 6+ digits at N=14. v1.1 (2026-08-22): Complete self-adjointness proof (23 pages). The paper previously uploaded a 14-page version; this version contains the full Carleman criterion argument and expanded numerical tables. Note: The construction is conditional on RH; no claim of proving RH is made.