Riemannium

The zeros of the Riemann zeta function are objects of pure arithmetic. They encode how the prime numbers are distributed, and nothing about them has anything to do with physics. The eigenvalues of a random Hermitian matrix are objects of pure probability, invented to model atomic nuclei too complicated to solve. There is no reason on earth for the two to be related.

They have the same statistics. Not approximately, not in some loose qualitative sense — the spacings between zeta zeros follow the distribution of eigenvalue spacings of large random Hermitian matrices, and specifically the unitary class rather than the orthogonal one. This was noticed in a common room over tea, has been checked numerically to punishing precision, and remains completely unexplained.

Before comparing anything we need zeros, and a great many of them. Zeta takes complex arguments, so its zeros are not something a graph crosses — but there is a standard trick. The Hardy Z function is real whenever zeta is evaluated on the critical line, and it vanishes exactly where zeta does. Finding zeros becomes watching an ordinary wiggly curve change sign.

Try it: The zeros here are computed as you drag, not looked up. While the window sits low enough to overlap the published list, the demo reports its own error against it. Slide upward and watch the zeros crowd together — the mean gap shrinks like one over the logarithm of the height, which is exactly the quantity the next comparison has to divide out.

Now put them side by side. On one side, gaps between zeta zeros. On the other, gaps between eigenvalues of random Hermitian matrices. Both divided by their own mean spacing so the comparison is about shape rather than scale. Neither set of numbers is stored anywhere — the zeros are found by the method above and the eigenvalues come from matrices generated when the page loads.

Try it: Move up in height and watch the amber bars settle onto the curve. The fit is good but not perfect low down and improves as you climb — which is the honest situation. This is a statement about a limit, and Odlyzko had to reach zeros near the 10²⁰-th before the agreement became visually exact.

Spacing between neighbours is only part of the story. Pair correlation asks the fuller question: given a zero, how much more or less likely is another zero at distance u than chance would suggest? Scattered points give the answer 1 at every distance — no separation is preferred. What Montgomery derived instead was 1 − (sin πu / πu)², which vanishes at the origin and then oscillates in toward 1. Dyson knew that function on sight.

The discovery has an unusually specific setting. It was not the product of a research programme aimed at connecting the two fields, because no such programme existed. It was a conversation between two people who had not met, one of whom happened to recognise the other’s formula.

It is worth being precise about what has been established. That the zeros and the eigenvalues share statistics is supported by an enormous weight of computation and by theorems covering restricted cases. What does not exist is a reason. Nobody has produced an argument deriving the spacing of the zeros from the arithmetic of the primes and arriving at the random matrix answer, and nobody has produced the operator whose eigenvalues the zeros would be. The closing lesson is about that gap. For the zeta function itself, and the explicit formula linking its zeros to the primes, see Number Theory.