The Riemann Hypothesis remains one of the most significant unsolved problems in mathematics. Apromising avenue for its resolution lies in the "Spectral Interpretation," which posits that the non-trivialzeros of the Riemann zeta function correspond to the eigenvalues of a quantum chaotic Hamiltonian.This paper presents a numerical verification of the Berry-Keating conjecture by simulating theeigenvalues of Random Unitary Matrices from the Circular Unitary Ensemble (CUE). We demonstratethat the nearest-neighbor spacing distribution of the normalized zeros exhibits level repulsionconsistent with the Wigner Surmise, deviating significantly from the Poisson distribution expected ofrandom numbers. Furthermore, we employ a semiclassical approximation of the Hardy Z-function tovisually reconstruct the "music of primes," confirming the spectral duality between prime numbers andzeta zeros.
Pao-Hsin Huang (Sun,) studied this question.