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Sunday, May 5, 2013

Riemann, Dyson, Montgomery, Quasi-crystals, Matrix Theory...


....and, quantum physics.

Below a longish and deep read from the Institute For Advanced Study on the mysterious linkage between the Riemann zeta function and physics (a lot to chew on or just try to comprehend):

https://www.ias.edu/about/publications/ias-letter/articles/2013-spring/primes-random-matrices


Just a couple of passages from the long piece:
 “ 'The Riemann zeta function remains one of the mysteries of modern mathematics. It is a function that we understand a lot about except for the most important question,' says Peter Sarnak, Professor in the School of Mathematics. 'It connects the theory of prime numbers, or encodes deep information about the theory of prime numbers, with the zeros. It controls the prime numbers in a way that nothing else we know does. While understanding prime numbers is an important problem, it is the generalizations of the Riemann zeta function and the objects associated with these that make it more significant.' ”....

"Quasi-crystals were discovered in 1984 and exist in spaces of one, two, or three dimensions. Dyson suggests mathematicians obtain a complete enumeration and classification of all one-dimensional quasi-crystals, the most prevalent type, with the aim of identifying one with a spectrum that corresponds to the Riemann zeta function and one that corresponds to the L-functions that resemble the Riemann zeta function. If it can be proved that a one-dimensional quasi-crystal has properties that identify it with the zeros of the Riemann zeta function, then the Riemann Hypothesis will have been proved."
I haven't had occasion to read the 2nd and 3rd volumes of Matthew Watkin's trilogy on prime numbers, but I can't help but think that the above article may overlap with or relate to some of the notions reached by Watkins who also writes of the 'vibrational' nature of the number system.


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