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The Spectrum of Riemannium

Brian Hayes

The year: 1972. The scene: Afternoon tea in Fuld Hall at the Institute for Advanced Study. The camera pans around the Common Room, passing by several Princetonians in tweeds and corduroys, then zooms in on Hugh Montgomery, boyish Midwestern number theorist with sideburns. He has just been introduced to Freeman Dyson, dapper British physicist.

Dyson: So tell me, Montgomery, what have you been up to?
Montgomery: Well, lately I've been looking into the distribution of the zeros of the Riemann zeta function.
Dyson: Yes? And?
Montgomery: It seems the two-point correlations go as.... (turning to write on a nearby blackboard):

Click to Enlarge Image

Dyson: Extraordinary! Do you realize that's the pair-correlation function for the eigenvalues of a random Hermitian matrix? It's also a model of the energy levels in a heavy nucleus—say U-238.

I present this anecdote in cinematic form because I expect to see it on the big screen someday, now that mathematicians outgun cowboys and secret agents at the box office. Besides, the screenplay genre gives me license to dramatize and embellish a little. By the time the movie opens at your local multiplex, the script doctors will have taken further liberties with the facts. For example, the equation for nuclear energy levels will have become the secret formula of the atomic bomb.

Even without Hollywood hyperbole, however, the chance encounter of Montgomery and Dyson was a genuinely dramatic moment. Their conversation revealed an unsuspected connection between areas of mathematics and physics that had seemed remote. Why should the same equation describe both the structure of an atomic nucleus and a sequence at the heart of number theory? And what do random matrices have to do with either of those realms? In recent years, the plot has thickened further, as random matrices have turned up in other unlikely places, such as games of solitaire, one-dimensional gases and chaotic quantum systems. Is it all just a cosmic coincidence, or is there something going on behind the scenes?

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