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Hugh Montgomery investigated and found that the statistical distribution of the zeros on the critical line has a certain property. The zeros tend not to be too closely together, but to repel. Visiting at the Institute for Advanced Study he showed this result to Freeman Dyson, one of the founders of the theory of random matrices, which is of importance in physics due to the fact that the eigenstates of a Hamiltonian, for example the energy levels of an atomic nucleus, satisfy such statistics. |
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Hugh Montgomery investigated and found that the statistical distribution of the zeros on the critical line has a certain property. The zeros tend not to be too closely together, but to repel. Visiting at the Institute for Advanced Study in 1972, he showed this result to Freeman Dyson, one of the founders of the theory of random matrices, which is of importance in physics due to the fact that the eigenstates of a Hamiltonian, for example the energy levels of an atomic nucleus, satisfy such statistics. |
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Dyson saw that the statistical distribution found by Montgomery was exactly the same as the pair correlation distribution for the eigenvalues of a random Hermitian matrix. Subsequent work has strongly born out this discovery, and the distribution of the zeros of the Riemann zeta function is now believed to satisfy the same statistics as the zeros of a random Hermitian matrix, the statistics of the so-called Gaussian Unitary Ensemble. Thus the conjecture of Polya and Hilbert now has a more solid basis, though it has not led to a proof of the Riemann hypothesis. |
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Dyson saw that the statistical distribution found by Montgomery was exactly the same as the pair correlation distribution for the eigenvalues of a random Hermitian matrix. Subsequent work has strongly born out this discovery, and the distribution of the zeros of the Riemann zeta function is now believed to satisfy the same statistics as the zeros of a random Hermitian matrix, the statistics of the so-called Gaussian Unitary Ensemble. Thus the conjecture of Polya and Hilbert now has a more solid basis, though it has not yet led to a proof of the Riemann hypothesis. |
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