Empirical validation of Montgomery's Pair Correlation Conjecture and Quantum Chaos
The normalized nearest-neighbor spacing between non-trivial zeros of the Riemann zeta function $\zeta(s)$ on the critical line $\text{Re}(s) = 1/2$ obeys the Gaussian Unitary Ensemble (GUE) distribution from random matrix theory. Hugh Montgomery's Pair Correlation Conjecture established that local zero gaps mirror the energy level repulsions observed in heavy atomic nuclei and quantum chaotic systems.
Unlike uncorrelated Poisson random variables—which produce exponential spacing distributions $P(s) = e^{-s}$ with frequent small gaps—GUE eigenvalues exhibit level repulsion ($P(s) \to 0$ as $s \to 0$). The theoretical curve is accurately modeled by the Wigner Surmise $P(s) = \frac{32}{\pi^2} s^2 e^{-\frac{4}{\pi} s^2}$.
This interactive visual compares empirical zero/eigenvalue spacings against both the GUE Wigner Surmise curve and Poisson noise. It demonstrates how spectral repulsion governs high-frequency zero distributions, reinforcing the deep physical duality between prime sieves and quantum operator spectra.