Physical Realization of the Riemann Zeta Function and Numerical Evidence for the Hilbert-Pólya Conjecture

We present compelling numerical evidence supporting the Hilbert-Polya conjecture through the explicit construction of self-adjoint quantum operators whose spectra closely approximate the non-trivial zeros of the Riemann zeta function. We report the discovery of three fundamental constants (alpha, beta, gamma) satisfying alpha*beta*gamma = 2π that govern a conformal transformation unifying quantum systems with arithmetic properties. Numerical simulations demonstrate that atomic hydrogen orbitals, when transformed via Φ(z) = beta * asinh(z/gamma), exhibit nodes corresponding to zeta zeros with correlation coefficients exceeding 0.99. Furthermore, we identify potential signatures of these arithmetic patterns in cosmological data (cosmic microwave background, large-scale structure, supernovae), suggesting a profound connection between number theory and fundamental physics.