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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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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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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.
Quantum chaos, random matrix theory, and the Riemann ζfunction
, 2010
"... Hilbert and Pólya put forward the idea that the zeros of the Riemann zeta function may have a spectral origin: the values of tn such that 1 2 + itn is a non trivial zero of ζ might be the eigenvalues of a selfadjoint operator. This would imply the Riemann Hypothesis. From the perspective of Physics ..."
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Cited by 4 (0 self)
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Hilbert and Pólya put forward the idea that the zeros of the Riemann zeta function may have a spectral origin: the values of tn such that 1 2 + itn is a non trivial zero of ζ might be the eigenvalues of a selfadjoint operator. This would imply the Riemann Hypothesis. From the perspective of Physics one might go further and consider the possibility that the operator in question corresponds to the quantization of a classical dynamical system. The first significant evidence in support of this spectral interpretation of the Riemann zeros emerged in the 1950’s in the form of the resemblance between the Selberg trace formula, which relates the eigenvalues of the Laplacian and the closed geodesics of a Riemann surface, and the Weil explicit formula in number theory, which relates the Riemann zeros to the primes. More generally, the Weil explicit formula resembles very closely a general class of Trace Formulae, written down by Gutzwiller, that relate quantum energy levels to classical periodic orbits in chaotic Hamiltonian systems. The second
Spectral Statistics in Chaotic Systems With a Point Interaction
"... We consider quantum systems with a chaotic classical limit that are perturbed by a pointlike scatterer. The spectral form factor K() for these systems is evaluated semiclassically in terms of periodic and diffractive orbits. It is shown for order _2 and 93 that offdiagonal contributions to the f ..."
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We consider quantum systems with a chaotic classical limit that are perturbed by a pointlike scatterer. The spectral form factor K() for these systems is evaluated semiclassically in terms of periodic and diffractive orbits. It is shown for order _2 and 93 that offdiagonal contributions to the form factor which involve diffractive orbits cancel exactly the diagonal contributions from diffractive orbits, implying that the perturbation by the scatterer does not change the spectral statistic. We further show that parametric spectral statistics for these systems are universal for small changes of the strength of the scatterer.
The quantum threedimensional Sinai . . .
, 1999
"... We present a comprehensive semiclassical investigation of the threedimensional Sinai billiard, addressing a few outstanding problems in “quantum chaos”. We were mainly concerned with the accuracy of the semiclassical trace formula in two and higher dimensions and its ability to explain the universa ..."
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We present a comprehensive semiclassical investigation of the threedimensional Sinai billiard, addressing a few outstanding problems in “quantum chaos”. We were mainly concerned with the accuracy of the semiclassical trace formula in two and higher dimensions and its ability to explain the universal spectral statistics observed in quantized chaotic systems. For this purpose we developed an efficient KKR algorithm to compute an extensive and accurate set of quantal eigenvalues. We also constructed a systematic method to compute millions of periodic orbits in a reasonable time. Introducing a proper measure for the semiclassical error and using the quantum and the classical databases for the Sinai billiards in two and three dimensions, we concluded that the semiclassical error (measured in units of the mean level spacing) is independent of the dimensionality, and diverges at most as log �. This is in contrast with previous estimates. The classical spectrum of lengths of periodic orbits was studied and shown to