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in the Popular Press

"The Spectrum of Riemannium," by Brian Hayes. American Scientist, July/August 2003, pages 296-300.

This article discusses the tantalizing affinities that have been discovered between the Riemann Hypothesis, random matrix theory, and particle physics. The Riemann Hypothesis, one of the outstanding problems in mathematics today, makes a conjecture about where a certain function, the Riemann zeta function, has the value of zero. The reason this problem has been considered so important is that its resolution would reveal fundamental information about that most complicated and mysterious set of numbers, the primes. And yet, since it was first proposed 150 years ago, the Riemann Hypothesis has steadfastly mathematicians' best efforts.

Random matrix theory studies the behavior of square arrays of random numbers that exhibit certain symmetries. It turns out that random matrix theory provides a good model for understanding the spectrum of energy levels of certain heavy particles. Even more surprising, this same model seems to predict with extraordinary accuracy the distribution of the zeros of the Riemann zeta function. The predictions have been carried out statistically by computer and explored in billions of cases, but a once-and-for-all mathematical proof that the model really fits has yet to be found.

"Is it all just a fluke, this apparent link between matrix eigenvalues, nuclear physics and zeta zeros?" Hayes asks in the article. "It could be, although a universe with such chance coincidences in its fabric might be considered even stranger than one with mysterious causal connections."

--- Allyn Jackson