The non-trivial zeros of ζ(s) lie on the critical line ½+it. Their spacing statistics match GUE (Gaussian Unitary Ensemble) random matrices — the Montgomery-Odlyzko law (1972/1987).
Controls
Mean spacing: —
First zero: t₁ ≈ 14.135
Montgomery's Pair Correlation
1 − (sin πu/πu)² — identical to GUE pair correlation.
Odlyzko (1987): computed 10⁷ zeros near t≈10²⁰, perfect GUE fit.
Wigner surmise for GUE: p(s) = (32/π²)s²e^(−4s²/π)
Connection: zeros = eigenvalues of a random Hermitian matrix. No proof yet.