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30
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
Random matrix theory and the derivative of the Riemann zeta function
, 2000
"... Random matrix theory (RMT) is used to model the asymptotics of the discrete moments of the derivative of the Riemann zeta function, ? (s), evaluated at the complex zeros + iγn, using the methods introduced by Keating and Snaith in [14]. We also discuss the probability distribution of ln ? ´(1/2 + ..."
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Cited by 34 (7 self)
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Random matrix theory (RMT) is used to model the asymptotics of the discrete moments of the derivative of the Riemann zeta function, ? (s), evaluated at the complex zeros + iγn, using the methods introduced by Keating and Snaith in [14]. We also discuss the probability distribution of ln ? ´(1/2 + iγn), proving the central limit theorem for the corresponding random matrix distribution and analysing its large deviations.
An elementary problem equivalent to the Riemann hypothesis
 Amer. Math. Monthly
"... ABSTRACT. The problem is: Let Hn = n∑ n ≥ 1, that with equality only for n = 1. j=1 1 j d ≤ Hn + exp(Hn)log(Hn), ..."
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Cited by 21 (2 self)
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ABSTRACT. The problem is: Let Hn = n∑ n ≥ 1, that with equality only for n = 1. j=1 1 j d ≤ Hn + exp(Hn)log(Hn),
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 20 (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.
Spectra, Pseudospectra, and Localization for Random Bidiagonal Matrices
 Comm. Pure Appl. Math
"... There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random nonhermitian periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a \bubble with wings" in the complex plane, and the corresponding ..."
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Cited by 13 (4 self)
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There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random nonhermitian periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a \bubble with wings" in the complex plane, and the corresponding eigenvectors are localized in the wings, delocalized in the bubble. Here, in addition to eigenvalues, pseudospectra are analyzed, making it possible to treat the nonperiodic analogues of these random matrix problems. Inside the bubble, the resolvent norm grows exponentially with the dimension. Outside, it grows subexponentially in a bounded region that is the spectrum of the in nitedimensional operator. Localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase, respectively, with distance from the diagonal. This article presents theorems that characterize the spectra, pseudospectra, and numerical range for the four cases of nite bidiagonal matrices, in nite bidiagonal matrices (\stochastic Toeplitz operators"), nite periodic matrices, and doubly in nite bidiagonal matrices (\stochastic Laurent operators").
APPLICATIONS OF THE LFUNCTIONS RATIOS CONJECTURES
 PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
, 2006
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Between classical and quantum
, 2005
"... The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, inclu ..."
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Cited by 12 (3 self)
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The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is Heisenberg’s ‘quantumtheoretical Umdeutung (reinterpretation) of classical observables’, which lies at the basis of quantization theory. Similarly, Bohr’s correspondence principle (in somewhat revised form) and Schrödinger’s wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from quantum mechanics. On the other hand, no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail. On the assumption that quantum mechanics is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely
Computed eigenmodes of planar regions
 IN RECENT ADVANCES IN DIFFERENTIAL EQUATIONS AND MATHEMATICAL PHYSICS, VOLUME 412 OF CONTEMP. MATH
, 2006
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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
Selberg’s trace formula: an introduction
 Proceedings of the International School ”Quantum Chaos on Hyperbolic Manifolds” (Schloss Reisensburg, Gunzburg
"... The aim of this short lecture course is to develop Selberg’s trace formula for a compact hyperbolic surface M, and discuss some of its applications. The main motivation for our studies is quantum chaos: the LaplaceBeltrami operator − ∆ on the surface M represents the quantum Hamiltonian of a partic ..."
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Cited by 3 (0 self)
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The aim of this short lecture course is to develop Selberg’s trace formula for a compact hyperbolic surface M, and discuss some of its applications. The main motivation for our studies is quantum chaos: the LaplaceBeltrami operator − ∆ on the surface M represents the quantum Hamiltonian of a particle, whose classical dynamics is governed by the (strongly chaotic) geodesic flow on the unit tangent bundle of M. The trace formula is currently the only available tool to analyze the fine structure of the spectrum of −∆; no individual formulas for its eigenvalues are known. In the case of more general quantum systems, the role of Selberg’s formula is taken over by the semiclassical Gutzwiller trace formula [10], [7]. We begin by reviewing the trace formulas for the simplest compact manifolds, the circle S 1 (Section 1) and the sphere S 2 (Section 2). In both cases, the corresponding geodesic flow is integrable, and the trace formula is a consequence of the Poisson summation formula. In the remaining sections we shall discuss the following topics: the Laplacian on the hyperbolic plane and isometries (Section 3); Green’s functions (Section 4); Selberg’s point pair invariants (Section 5); The ghost of the sphere (Section 6); Linear operators on hyperbolic surfaces (Section 7); A trace formula for hyperbolic cylinders and poles of the scattering matrix (Section 8); Back to general hyperbolic surfaces (Section 9); The spectrum of a compact surface, Selberg’s pretrace and trace formulas (Section 10); Heat kernel and Weyl’s law (Section 11); Density of closed geodesics (Section 12); Trace of the resolvent (Section 13); Selberg’s zeta function (Section 14); Suggestions for exercises and further reading (Section 15). Our main references are Hejhal’s classic lecture notes [12, Chapters one and two], Balazs and Voros ’ excellent introduction [1], and Cartier and Voros’ nouvelle interprétation [6]. Section 15 comprises a list of references for further reading. These notes are based on lectures given at the International School Quantum