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16
Zeroes of Zeta Functions and Symmetry
, 1999
"... Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the “function field” analogues, that is zeta functions of cur ..."
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Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the “function field” analogues, that is zeta functions of curves over finite fields and their generalizations. For these a spectral interpretation for their zeroes exists in terms of eigenvalues of Frobenius on cohomology. Secondly, the developments, both theoretical and numerical, on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection. Moreover, the lowlying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups. In this paper we review these developments. In order to present the material fluently, we do not proceed in chronological order of discovery. Also, in concentrating entirely on the subject matter of the title, we are ignoring the standard body of important work that has been done on the zeta function and Lfunctions.
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 52 (10 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.
From endomorphisms to automorphisms and back: dilations and full corners
 J. London Math. Soc
"... Abstract. When S is a discrete subsemigroup of a discrete group G such that G = S −1 S, it is possible to extend circlevalued multipliers from S to G; to dilate (projective) isometric representations of S to (projective) unitary representations of G; and to dilate/extend actions of S by injective e ..."
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Abstract. When S is a discrete subsemigroup of a discrete group G such that G = S −1 S, it is possible to extend circlevalued multipliers from S to G; to dilate (projective) isometric representations of S to (projective) unitary representations of G; and to dilate/extend actions of S by injective endomorphisms of a C*algebra to actions of G by automorphisms of a larger C*algebra. These dilations are unique provided they satisfy a minimality condition. The (twisted) semigroup crossed product corresponding to an action of S is isomorphic to a full corner in the (twisted) crossed product by the dilated action of G. This shows that crossed products by semigroup actions are Morita equivalent to crossed products by group actions, making powerful tools available to study their ideal structure and representation theory. The dilation of the system giving the Bost– Connes Hecke C*algebra from number theory is constructed explicitly as an application: it is the crossed product C0(Af) ⋊ Q ∗ +, corresponding to the multiplicative action of the positive rationals on the additive group Af of finite adeles.
The numbertheoretical spin chain and the Riemann zeroes
 Comm. Math. Phys
, 1998
"... Abstract It is an empirical observation that the Riemann zeta function can be well approximated in its critical strip using the NumberTheoretical Spin Chain. A proof of this would imply the Riemann Hypothesis. Here we relate that question to the one of spectral radii of a family of Markov chains. T ..."
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Cited by 9 (1 self)
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Abstract It is an empirical observation that the Riemann zeta function can be well approximated in its critical strip using the NumberTheoretical Spin Chain. A proof of this would imply the Riemann Hypothesis. Here we relate that question to the one of spectral radii of a family of Markov chains. This in turn leads to the question whether certain graphs are Ramanujan. The general idea is to explain the pseudorandom features of certain numbertheoretical functions by considering them as observables of a spin chain of statistical mechanics. In an Appendix we relate the free energy of that chain to the Lewis Equation of modular theory. 1 Introduction The Euler
Number Theory, Dynamical Systems and Statistical Mechanics
, 1998
"... We shortly review recent work interpreting the quotient ζ(s − 1)/ζ(s) of Riemann zeta functions as a dynamical zeta function. The corresponding interaction function (Fourier transform of the energy) has been shown to be ferromagnetic, i.e. positive. On the additive group we set inductively Gk: = (Z/ ..."
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Cited by 8 (2 self)
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We shortly review recent work interpreting the quotient ζ(s − 1)/ζ(s) of Riemann zeta functions as a dynamical zeta function. The corresponding interaction function (Fourier transform of the energy) has been shown to be ferromagnetic, i.e. positive. On the additive group we set inductively Gk: = (Z/2Z) k, with Z/2Z = ({0, 1}, +). h0: = 1, hk+1(σ, 0): = hk(σ) and hk+1(σ, 1): = hk(σ) + hk(1 − σ), (1) where σ = (σ1,..., σk) ∈ Gk and 1 − σ: = (1 − σ1,..., 1 − σk) is the inverted configuration. The sequences hk(σ) of integers, written in lexicographic order, coincide with the denominators of the modified Farey sequence. We now formally interpret σ ∈ Gk as a configuration of a spin chain with k spins and energy function Hk: = ln(hk). Thus we may interpret
A lower bound in an approximation problem involving the zeros of the Riemann zeeta function
, 2001
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Weyls Law: Spectral Properties of the Laplacian in Mathematics and Physics
 Mathematical Analysis of Evolution, Information and Complexity. WileyVerlag, 2009. 27 Rajendra Bhatia, Linear Algebra to Quantum Cohomology: The Story of Alfred Horn’s Inequalities, Amer. Math. Monthly 108(2001
"... Weyl’s law is in its simplest version a statement on the asymptotic growth of the eigenvalues of the Laplacian on bounded domains with Dirichlet and Neumann boundary conditions. In the typical applications in physics one deals either with the Helmholtz wave equation describing the vibrations of a st ..."
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Cited by 5 (0 self)
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Weyl’s law is in its simplest version a statement on the asymptotic growth of the eigenvalues of the Laplacian on bounded domains with Dirichlet and Neumann boundary conditions. In the typical applications in physics one deals either with the Helmholtz wave equation describing the vibrations of a string, a membrane
padic DifferenceDifference LotkaVolterra Equation and UltraDiscrete
, 2001
"... Abstract. In this article, we have studied the differencedifference LotkaVolterra equations in padic number space and its padic valuation version. We pointed out that the structure of the space given by taking the ultradiscrete limit is the same as that of the padic valuation space. ..."
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Cited by 3 (3 self)
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Abstract. In this article, we have studied the differencedifference LotkaVolterra equations in padic number space and its padic valuation version. We pointed out that the structure of the space given by taking the ultradiscrete limit is the same as that of the padic valuation space.
On Fourier and Zeta(s)
, 2002
"... This is the final version for FORUM MATHEMATICUM; October 2002 We study some of the interactions between the Fourier Transform and the Riemann zeta ..."
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This is the final version for FORUM MATHEMATICUM; October 2002 We study some of the interactions between the Fourier Transform and the Riemann zeta
The BerryKeating Operator on compact quantum graphs with general selfadjoint realizations. Ulmer Seminare 2009
"... Abstract. The BerryKeating operator HBK: = −i~ xddx + ..."
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Abstract. The BerryKeating operator HBK: = −i~ xddx +