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Random Matrix Theory and ζ(1/2 + it)
, 2000
"... We study the characteristic polynomials Z(U,#)of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of and Z/Z # , and from these we obtain the asymptotics of the value distributions and cumulants of the re ..."
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Cited by 85 (15 self)
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We study the characteristic polynomials Z(U,#)of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of and Z/Z # , and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log Z as N ##. In the
The Riemann Zeros and Eigenvalue Asymptotics
 SIAM Rev
, 1999
"... Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many feat ..."
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Cited by 42 (5 self)
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Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display randommatrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The RiemannSiegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.
Random matrices and Lfunctions
 J. PHYS A MATH GEN
, 2003
"... In recent years there has been a growing interest in connections between the statistical properties of number theoretical Lfunctions and random matrix theory. We review the history of these connections, some of the major achievements and a number of applications. ..."
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Cited by 19 (7 self)
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In recent years there has been a growing interest in connections between the statistical properties of number theoretical Lfunctions and random matrix theory. We review the history of these connections, some of the major achievements and a number of applications.
2003 Random matrix theory and the zeros of ζ
 Preprint mathph/0207044
"... Abstract. We study the density of the roots of the derivative of the characteristic polynomial Z(U, z) of an N × N random unitary matrix with distribution given by Haar measure on the unitary group. Based on previous random matrix theory models of the Riemann zeta function ζ(s), this is expected to ..."
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Cited by 10 (0 self)
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Abstract. We study the density of the roots of the derivative of the characteristic polynomial Z(U, z) of an N × N random unitary matrix with distribution given by Haar measure on the unitary group. Based on previous random matrix theory models of the Riemann zeta function ζ(s), this is expected to be an accurate description for the horizontal distribution of the zeros of ζ ′ (s) to the right of the critical line. We show that as N → ∞ the fraction of roots of Z ′ (U, z) that lie in the region 1−x/(N −1) ≤ z  < 1 tends to a limit function. We derive asymptotic expressions for this function in the limits x → ∞ and x → 0 and compare them with numerical experiments. Mathematics Subject Classification: 15A52, 11M99Random matrix theory and the zeros of ζ ′ (s) 2 1.
Correlations for pairs of closed geodesics
 Invent. Math
"... Abstract. In this article we consider natural counting problems for closed geodesics on negatively curved surfaces. We present asymptotic estimates for pairs of closed geodesics, the differences of whose lengths lie in a prescribed family of shrinking intervals. Related pair correlation problems hav ..."
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Cited by 9 (2 self)
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Abstract. In this article we consider natural counting problems for closed geodesics on negatively curved surfaces. We present asymptotic estimates for pairs of closed geodesics, the differences of whose lengths lie in a prescribed family of shrinking intervals. Related pair correlation problems have been studied in both Quantum Chaos and number theory. One of the most striking properties of negatively curved surfaces is the regularity of the distribution of the lengths of their closed geodesics. This is shown by the wellknown prime geodesic theorem. More precisely, let V denote a compact surface with a C ∞ Riemannian metric of strictly negative curvature. Given any closed geodesic
Triple correlation of the Riemann zeros
"... We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function [11] to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested by Bogomolny and Keating [6] taking inspiration from semi ..."
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Cited by 4 (2 self)
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We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function [11] to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested by Bogomolny and Keating [6] taking inspiration from semiclassical methods. At that point they did not write out the answer explicitly, so we do that here, illustrating that by our method all the lower order terms down to the constant can be calculated rigourously if one assumes the ratios conjecture of Conrey, Farmer and Zirnbauer. Bogomolny and Keating [4] returned to their previous results simultaneously with this current work, and have written out the full expression. The result presented in this paper agrees precisely with their formula, as well as with our numerical computations, which we include here. We also include an alternate proof of the triple correlation of eigenvalues from random U(N) matrices which follows a nearly identical method to that for the Riemann zeros, but is based on
Twopoint correlations of the Gaussian symplectic ensemble from periodic orbits
, 2000
"... . We determine the asymptotics of the twopoint correlation function for quantum systems with halfinteger spin which show chaotic behaviour in the classical limit using a method introduced by Bogomolny and Keating [Phys. Rev. Lett. 77 (1996) 14721475]. For timereversal invariant systems we obta ..."
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Cited by 1 (0 self)
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. We determine the asymptotics of the twopoint correlation function for quantum systems with halfinteger spin which show chaotic behaviour in the classical limit using a method introduced by Bogomolny and Keating [Phys. Rev. Lett. 77 (1996) 14721475]. For timereversal invariant systems we obtain the leading terms of the twopoint correlation function of the Gaussian symplectic ensemble. Special attention has to be paid to the role of Kramers' degeneracy. PACS numbers: 03.65.Sq, 05.45.Mt z Email address: kep@physik.uniulm.de Twopoint correlations of the GSE from periodic orbits 2 Understanding correlations of energy levels of quantum mechanical systems whose classical limit exhibits chaotic motion is one of the major topics in quantum chaos. The bridge between quantum mechanics and classical mechanics is provided by the Gutzwiller trace formula [1] which relates the quantum mechanical density of states d(E) = P n ffi(E \Gamma E n ) to a sum over periodic orbits of the cor...
Angular SelfIntersections For Closed Geodesics On Surfaces
"... In this note we consider asymptotic results for selfinterections of closed geodesics on surfaces for which the angle of the intersection occurs in a given arc. ..."
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Cited by 1 (0 self)
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In this note we consider asymptotic results for selfinterections of closed geodesics on surfaces for which the angle of the intersection occurs in a given arc.
Random matrix theory and the zeros of ζ'(s)
 PREPRINT MATHPH/0207044
, 2002
"... We study the density of the roots of the derivative of the characteristic polynomial Z(U, z) of an N × N random unitary matrix with distribution given by Haar measure on the unitary group. Based on previous random matrix theory models of the Riemann zeta function ζ(s), this is expected to be an accu ..."
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We study the density of the roots of the derivative of the characteristic polynomial Z(U, z) of an N × N random unitary matrix with distribution given by Haar measure on the unitary group. Based on previous random matrix theory models of the Riemann zeta function ζ(s), this is expected to be an accurate description for the horizontal distribution of the zeros of ζ ′ (s) to the right of the critical line. We show that as N → ∞ the fraction of roots of Z ′ (U, z) that lie in the region 1−x/(N −1) ≤ z  < 1 tends to a limit function. We derive asymptotic expressions for this function in the limits x → ∞ and x → 0 and compare them with numerical experiments.