A handbook of terms used in chaos and quantum chaos

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Odlyzko Journal: Math. Abstract: A numerical study of the distribution of spacings between zeros of the Riemann zeta function is presented.

A handbook of terms used in chaos and quantum chaos

It is based on values for the first zeros and for zeros number to that are accurate to within , and which were calculated on the Cray-1 and Cray X-MP computers. This study tests the Montgomery pair correlation conjecture as well as some further conjectures that predict that the zeros of the zeta function behave like eigenvalues of random Hermitian matrices. Matrices of this type are used in modeling energy levels in physics, and many statistical properties of their eigenvalues are known. The agreement between actual statistics for zeros of the zeta function and conjectured results is generally good, and improves at larger heights.

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Several initially unexpected phenomena were found in the data and some were explained by relating them to the primes. References [Enhancements On Off] What's this? Berry , Semiclassical theory of spectral rigidity , Proc. London Ser. A , no. MR [3] M. Berry, "Riemann's zeta function: A model for quantum chaos? Second Internat. Seligman, ed. To appear. Bohigas , M.

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Giannoni , and C. Schmit , Characterization of chaotic quantum spectra and universality of level fluctuation laws , Phys. Bohigas, M. Schmit, "Spectral properties of the Laplacian and random matrix theories," J. Bohigas, R. Pandey, "Higher-order correlations in spectra of complex systems," Phys. Hejhal, manuscript in preparation. Brent , Algorithms for minimization without derivatives , Prentice-Hall, Inc. Prentice-Hall Series in Automatic Computation.

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Brody , J. Flores , J. French , P. Mello , A. Pandey , and S. Wong , Random-matrix physics: spectrum and strength fluctuations , Rev. Modern Phys. Software , v. Chambers, W. Cleveland, B.

noisulecol.gq Mehta , Some asymptotic expressions for prolate spheroidal functions and for the eigenvalues of differential and integral equations of which they are solutions , J. Mathematical Phys. Mehta , Asymptotic behavior of spacing distributions for the eigenvalues of random matrices , J.

Conrey , A. Ghosh , D. Goldston , S. Gonek , and D. Heath-Brown , On the distribution of gaps between zeros of the zeta-function , Quart. Oxford Ser. Brent in Math. A , — Dyson , Statistical theory of the energy levels of complex systems. Gebiete 57 , no. I , Trans. Reine Angew. II , Trans.

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Gallagher , On the distribution of primes in short intervals , Mathematika 23 , no. Gallagher , Pair correlation of zeros of the zeta function , J. Gallagher and Julia H. Mueller , Primes and zeros in short intervals , J. MR [32] A. MR [33] A.

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Number Theory 17 , no. Goldston , Prime numbers and the pair correlation of zeros of the zeta-functions , Topics in analytic number theory Austin, Tex. Texas Press, Austin, TX, , pp. MR [35] D. Heath-Brown and D. Goldston , A note on the differences between consecutive primes , Math. Goldston and Hugh L.

Montgomery , Pair correlation of zeros and primes in short intervals , Analytic number theory and Diophantine problems Stillwater, OK, Progr. MR [37] S. MR [38] A. Guinand , A summation formula in the theory of prime numbers , Proc. London Math.

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Gutzwiller , Stochastic behavior in quantum scattering , Phys. Order in chaos Los Alamos, N. Abramowitz and I. Stegun, eds. Haq, A. Bohigas, "Fluctuation properties of nuclear energy levels: Do theory and experiment agree? Heath-Brown , Gaps between primes, and the pair correlation of zeros of the zeta function , Acta Arith.

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The theory of the Riemann zeta-function with applications. MR [44] D. Joyner, "Distribution theorems for L -functions. Joyner, "On the Dyson-Montgomery hypothesis. Karkoschka and P. Stuart, The Advanced Theory of Statistics , 3rd ed. Sherman Lehman , On the distribution of zeros of the Riemann zeta-function , Proc. Winter , On the zeros of the Riemann zeta function in the critical strip. IV , Math. MR [55] M.

Mehta and J. Pure Appl. MR [56] H. Montgomery , The pair correlation of zeros of the zeta function , Analytic number theory Proc. Pure Math. XXIV, St.