how M. Berry came to be involved with the Riemann Hypothesis

This is taken from Marcus du Sautoy's book The Music of the Primes (Fourth Estate, 2003) p.278:

"He became fascinated by prime numbers in the 1980's after reading an article in the Mathematical Intelligencer entitled 'The first 50 million prime numbers'...by Don Zagier [wherein he] described how the zeros in Riemann's landscape could be used to create waves which magically reproduced how many primes one can expect to find as one counts higher. 'It was a beautiful article. I thought the Riemann zeros were wonderful things.' Berry was taken with the very physical interpretation of Riemann's discovery - that there is music in the primes...

Berry's interest in the primes coincided with his growing understanding of the differences between the statistics of energy levels in electrons playing quantum billiards and the [spectra of random matrices]. 'I thought it might be interesting to look again at the story of the Riemann zeros and Dyson's ideas in the light of the new connections with quantum chaos.' Would the special statistics that Berry had discovered in the energy levels of quantum billiards be reflected in the statistics of the zeros of the Riemann zeta [function]? 'I thought it would be very nice to see if the zeros actually behaved in this way, and I did some rough calculations.' But he didnít have enough data. 'Then I heard of Odlyzko, who'd done these epic calculations. I wrote to him and he was wonderfully helpful. He explained to me that he'd been a little worried because his calculations beyond a certain point had started to show some deviations. He thought he must have made a mistake in his computations.'

But Odlyzko did not have the insights of a physicist. When Berry compared the zeros to the energy levels of chaotic quantum billiards, he found a perfect match. The discrepancies that Odlyzko had observed turned out to be the first sign of the difference between the statistics of frequencies in a random [matrix] and the energy levels of chaotic quantum billiards. He had not been aware of this new chaotic quantum system, but Berry recognised it straight away:

"This was a great moment because it was obviously right. This was to me absolute convincing circumstantial evidence that if you think the Riemann Hypothesis is true, then the Riemann zeros would have underlying them not just a quantum system, but a quantum system with a classical counterpart, moderately simple but chaotic. It was a lovely moment. That was, if you like, something that quantum mechanics provided for the theory of the Riemann zeros."

 


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