Holographic Realization of Riemann Zeros via a Hilbert-P´olya Operator

We construct a self-adjoint Schr¨odinger-type operator whose eigenvalues spec- trally align with segments of the imaginary parts of the nontrivial zeros of the Riemann zeta function, inspired by the Hilbert-P´olya conjecture. The potential is derived from the AdS/CFT correspondence via the boundary projection of bulk quasinormal modes (QNMs) in the BTZ black hole background. Using a finite- difference method, we compute low-lying eigenvalues with absolute errors between 1 31 × 10−4 and 5 27 × 10−3 for the first 10 Riemann zeros. The same spectral structure partially realizes the oscillatory behavior of the prime density function, achieving L2 errors of 0.0076 and 0.0037 for 20 and 30 spectral modes, respectively. This establishes a generative spectral duality, bridging spectral geometry, quan- tum gravity, and number theory...