Prime Number Lattice Operator and Gue Statistics: A Concrete Realization of the Hilbert–pólya Conjecture

Key Points

• This work aims to explore the relationship between prime number distribution and random matrix theory via the Hilbert-Pólya conjecture.
• Introduced a self-adjoint operator K̃ on the prime number lattice.
• Demonstrated GUE spectral statistics and calculated KS p-value.
• Proved rank-one decomposition with a closed-form secular equation.
• Provided numerical evidence for a non-vanishing spectral gap.
• Observed GUE spectral statistics align with Riemann zeros (KS p-value = 0.905).
• Established an exact rank-one decomposition relating to prime number distribution.
• Numerical evidence supports a spectral gap of approximately Δ∞ ≈ 0.961.

Abstract