The zeros of the Riemann zeta function keep their distance from one another. So do the energy levels of a uranium nucleus, and the eigenvalues of a large random matrix. In 1972, over tea at the Institute for Advanced Study, Hugh Montgomery and Freeman Dyson noticed that the zeros and the eigenvalues do it in exactly the same way. Half a century on, nobody knows why.
Live canvas visualisations are unavailable in this browser, so the strips and charts below are blank; the surrounding text still describes what they show.
Each strip shows consecutive levels, rescaled so the average gap is 1. The gold ticks are real zeros of ζ(½ + it), computed for this page. The teal ticks are eigenvalues of a freshly sampled random Hermitian matrix. The rose ticks are points scattered at random. Watch the gaps: two of the three almost never let a pair of ticks touch.
Collect the gap between each level and the next. Random points give gaps of every size, including many tiny ones — the curve e−s peaks at zero. Zeta zeros and matrix eigenvalues refuse small gaps: the curve is pinned to zero at s = 0 and rises like s². The page keeps sampling matrices as you watch; the gold histogram is the data.
Switch the block of zeros above and watch the gold bars. The first 20,000 zeros are slightly more rigid than GUE — their gap variance is 0.157 against GUE's 0.180 — a known correction that shrinks like 1/log t. By t = 10⁶ the variance has climbed to 0.166: most of the way to GUE, but not there yet. Convergence this slow is why Odlyzko eventually computed zeros near the 10²⁰th.
Montgomery did not compute spacings; he computed how often any two zeros sit at distance r. Assuming the Riemann hypothesis, and only over the range his method could reach, he got the density 1 − (sin πr / πr)² — and conjectured it held everywhere. Dyson recognised it instantly: the pair correlation of GUE eigenvalues, from his own work a decade earlier. The dip near zero is the repulsion. At these low heights the zeros are still more rigid than GUE — the same finite-height effect section II measures as a gap variance of 0.157 against 0.180 — so the gold bars sit visibly under the dashed curve near r = 0. The slow climb back to 1 says the zeros are spread more evenly than chance would ever arrange them.
Dyson's threefold way sorts systems into three symmetry classes, distinguished by one number β: how fast the probability of a small gap vanishes, p(s) ∝ sβ. β = 1 is the orthogonal ensemble (time-reversal symmetry), β = 2 the unitary ensemble, GUE (time-reversal symmetry broken — and, empirically, zeta), β = 4 the symplectic (time-reversal symmetry with half-integer spin). The dial below is continuous, using the Dumitriu–Edelman tridiagonal model, so you can watch repulsion switch on from nothing.
Three unlabeled spectra. One is a stretch of real zeta zeros, one is a random matrix, one is random points. Pick the zeros. The random points are easy to rule out — then you are left with the question the whole field is stuck on.
Random matrices: β-Hermite tridiagonal ensemble (Dumitriu & Edelman, 2002), eigenvalues by implicit QL, unfolded with the semicircle law — with the exact Gaussian count at β = 0, and an ensemble-averaged counting function at the smallest non-zero β, where the semicircle law does not yet hold — central 60% of each spectrum retained. The pair-correlation curve is the exact GUE form; the spacing curve, like the dial's curve, is the Wigner-type surmise normalised to unit mean, which differs from the exact (Gaudin) distribution by about a percent — its variance is 0.178 against the exact 0.180 quoted above. Background: Montgomery, The pair correlation of zeros of the zeta function (1973); Dyson's letter to Selberg of 7 April 1972, identifying Montgomery's density with the GUE pair correlation (reproduced in From Prime Numbers to Nuclear Physics and Beyond, IAS Institute Letter, Spring 2013); Dyson, Birds and Frogs (Notices of the AMS, 2009), for his own account of the tea-time conversation; Dyson, The threefold way (1962); Wigner (1951), on neutron resonance levels in heavy nuclei; Odlyzko, On the distribution of spacings between zeros of the zeta function (1987), and The 10²⁰-th zero of the Riemann zeta function and 175 million of its neighbors (1992, unpublished manuscript); Rudnick & Sarnak, Zeros of principal L-functions and random matrix theory (1996); Krbálek & Šeba, on the buses of Cuernavaca, J. Phys. A 33 (2000) L229; Mehta, Random Matrices.
Contact the author: np56123@icloud.com