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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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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
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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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
The Schrödinger Operator with Morse Potential on the Right Half Line
, 712
"... This paper studies the Schrödinger operator with Morse potential Vk(u) = 1 4 e2u + ke u on a right halfline [u0, ∞), and determines the Weyl asymptotics of eigenvalues for constant boundary conditions at the endpoint u0. In consequence it obtains information on the location of zeros of the Whittak ..."
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This paper studies the Schrödinger operator with Morse potential Vk(u) = 1 4 e2u + ke u on a right halfline [u0, ∞), and determines the Weyl asymptotics of eigenvalues for constant boundary conditions at the endpoint u0. In consequence it obtains information on the location of zeros of the Whittaker function Wκ,µ(x), for fixed real parameters κ,x with x> 0, viewed as an entire function in the complex variable µ. In this case all zeros lie on the imaginary axis, with the exception, if k < 0 of a finite number of real zeros which lie in the interval κ  < k. We obtain an asymptotic formula for the number of zeros N(T) = {ρ  Wκ,ρ(x) = 0, Im(ρ)  < T} of the form N(T) = 2 2 πT log T + π (2log 2−1−log x)T +O(1). Parallels are observed with zeros of the Riemann zeta function. 1.
RANDOM POLYNOMIALS, RANDOM MATRICES, AND LFUNCTIONS, II
"... Abstract. We show that the Circular Orthogonal Ensemble of random matrices arises naturally from a family of random polynomials. This sheds light on the appearance of random matrix statistics in the zeros of the Riemann zetafunction. 1. ..."
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Abstract. We show that the Circular Orthogonal Ensemble of random matrices arises naturally from a family of random polynomials. This sheds light on the appearance of random matrix statistics in the zeros of the Riemann zetafunction. 1.
THE ZETA FUNCTION ON THE CRITICAL LINE: NUMERICAL EVIDENCE FOR MOMENTS AND RANDOM MATRIX THEORY MODELS
"... Abstract. Results of extensive computations of moments of the Riemann zeta function on the critical line are presented. Calculated values are compared with predictions motivated by random matrix theory. The results can help in deciding between those and competing predictions. It is shown that for hi ..."
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Abstract. Results of extensive computations of moments of the Riemann zeta function on the critical line are presented. Calculated values are compared with predictions motivated by random matrix theory. The results can help in deciding between those and competing predictions. It is shown that for high moments and at large heights, the variability of moment values over adjacent intervals is substantial, even when those intervals are long, as long as a block containing 109 zeros near zero number 1023. More than anything else, the variability illustrates the limits of what one can learn about the zeta function from numerical evidence. It is shown the rate of decline of extreme values of the moments is modelled relatively well by power laws. Also, some long range correlations in the values of the second moment, as well as asymptotic oscillations in the values of the shifted fourth moment, are found. The computations described here relied on several representations of the zeta function. The numerical comparison of their effectiveness that is presented is of independent interest, for future large scale computations. 1.
An Elementary and Real Approach to Values of the Riemann Zeta Function ∗
, 812
"... An elementary approach for computing the values at negative integers of the Riemann zeta function is presented. The approach is based on a new method for ordering the integers and a new method for summation of divergent series. We show that the values of the Riemann zeta function can be computed, wi ..."
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An elementary approach for computing the values at negative integers of the Riemann zeta function is presented. The approach is based on a new method for ordering the integers and a new method for summation of divergent series. We show that the values of the Riemann zeta function can be computed, without using the theory of analytic continuation and functions of complex variable.
A RANDOM MATRIX MODEL FOR ELLIPTIC CURVE LFUNCTIONS OF FINITE CONDUCTOR
, 2011
"... Abstract. We propose a random matrix model for families of elliptic curve Lfunctions of finite conductor. A repulsion of the critical zeros of these Lfunctions away from the center of the critical strip was observed numerically by S. J. Miller in 2006 [50]; such behaviour deviates qualitatively fr ..."
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Abstract. We propose a random matrix model for families of elliptic curve Lfunctions of finite conductor. A repulsion of the critical zeros of these Lfunctions away from the center of the critical strip was observed numerically by S. J. Miller in 2006 [50]; such behaviour deviates qualitatively from the conjectural limiting distribution of the zeros (for large conductors this distribution is expected to approach the onelevel density of eigenvalues of orthogonal matrices after appropriate rescaling). Our purpose here is to provide a random matrix model for Miller’s surprising discovery. We consider the family of even quadratic twists of a given elliptic curve. The main ingredient in our model is a calculation of the eigenvalue distribution of random orthogonal matrices whose characteristic polynomials are larger than some given value at the symmetry point in the spectra. We call this subensemble of SO(2N) the excised orthogonal ensemble. The sievingoff of matrices with small values of the characteristic polynomial is akin to the discretization of the central values of Lfunctions implied by the formulæ of Waldspurger and KohnenZagier. The cutoff scale
PACIFIC JOURNAL OF MATHEMATICS Vol. 217, No. 2, 2004 OPERATORS AND DIVERGENT SERIES
"... We give a natural extension of the classical definition of Césaro convergence of a divergent sequence/function. This involves understanding the spectrum of eigenvalues and eigenvectors of a certain Césaro operator on a suitable space of functions or sequences. The essential idea is applicable in ide ..."
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We give a natural extension of the classical definition of Césaro convergence of a divergent sequence/function. This involves understanding the spectrum of eigenvalues and eigenvectors of a certain Césaro operator on a suitable space of functions or sequences. The essential idea is applicable in identical fashion to other summation methods such as Borel’s. As an example we show how to obtain the analytic continuation of the Riemann zeta function ζ(z) for Re z ≤ 1 directly from generalised Césaro summation of its divergent defining series. We discuss a variety of analytic and symmetry properties of these generalised methods and some possible further applications. 1.
mean spectral density as the Riemann zeros
, 2011
"... For the classical hamiltonian (x +1/x)(p +1/p), with position x and conjugate momentum p,allorbitsarebounded. Afterasymmetrization,thecorresponding quantum integral equation possesses a family of selfadjoint extensions: compact operators on the entire positive x axis, labelled by an angle α specify ..."
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For the classical hamiltonian (x +1/x)(p +1/p), with position x and conjugate momentum p,allorbitsarebounded. Afterasymmetrization,thecorresponding quantum integral equation possesses a family of selfadjoint extensions: compact operators on the entire positive x axis, labelled by an angle α specifying the boundary condition at the origin, with a discrete spectrum of real energies E.Onthecylinder{− ∞ < E < ∞,0 � α<2π},thereisasingleeigencurvein the form of a helix winding clockwise. The rise between successive windings gets sharper as the scaled Planck’s constant decreases. This behaviour can be understood semiclassically. The first two terms of the asymptotic eigenvalue density are the same as those for the density of heights of the Riemann zeros. PACS numbers: 02.30.Mv, 02.30.Tb, 03.65.Ge, 03.65.Sq (Some figures in this article are in colour only in the electronic version)
unknown title
, 2006
"... Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric αstable processes. ..."
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Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric αstable processes.