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25
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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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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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.
Some Analogies Between Number Theory and Dynamical Systems on Foliated Spaces
, 1998
"... . In this article we describe what a cohomology theory related to zeta and Lfunctions for algebraic schemes over the integers should look like. We then point out some striking analogies with the leafwise reduced cohomology of certain foliated dynamical systems. 1991 Mathematics Subject Classificati ..."
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Cited by 26 (5 self)
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. In this article we describe what a cohomology theory related to zeta and Lfunctions for algebraic schemes over the integers should look like. We then point out some striking analogies with the leafwise reduced cohomology of certain foliated dynamical systems. 1991 Mathematics Subject Classification: Primary 14A20; Secondary 14G10, 14F99, 58F18, 58F20. Keywords and Phrases: Lfunctions of motives, leafwise cohomology, dynamical systems, foliations. 1 Introduction For the arithmetic study of varieties over finite fields powerful cohomological methods are available which in particular shed much light on the nature of the corresponding zeta functions. These investigations culminated in Deligne's proof of an analogue of the Riemann conjecture for such zeta functions. This had been the hardest part of the Weil conjectures. For algebraic schemes over Spec Z and in particular for the Riemann zeta function no cohomology theory has yet been developed that could serve similar purposes. For a ...
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.
Noncommutative Geometry and the Riemann Zeta Function
 Mathematics: Frontiers and perspectives, IMU 2000
"... According to my first teacher Gustave Choquet one does, by openly facing a well known unsolved problem, run the risk of being remembered more by one’s failure than anything else. After reaching a certain age, I realized that waiting “safely ” until one reaches the endpoint of one’s life is an equal ..."
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Cited by 12 (4 self)
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According to my first teacher Gustave Choquet one does, by openly facing a well known unsolved problem, run the risk of being remembered more by one’s failure than anything else. After reaching a certain age, I realized that waiting “safely ” until one reaches the endpoint of one’s life is an equally selfdefeating alternative. In this paper I shall first look back at my early work on the classification of von Neumann algebras and cast it in the unusual light of André Weil’s Basic Number Theory. I shall then explain that this leads to a natural spectral interpretation of the zeros of the Riemann zeta function and a geometric framework in which the Frobenius, its eigenvalues and the Lefschetz formula interpretation of the explicit formulas continue to hold even for number fields. We shall then prove the positivity of the Weil distribution assuming the validity of the analogue of the Selberg trace formula. The latter remains unproved and is equivalent to RH for all Lfunctions with Grössencharakter. 1 Local class field theory and the classification of factors Let K be a local field, i.e. a nondiscrete locally compact field. The action of K ∗ = GL1(K) on the additive group K by multiplication, (1) (λ, x) → λx ∀ λ ∈ K ∗ , x ∈ K, together with the uniqueness, up to scale, of the Haar measure of the additive group K, yield a homomorphism, (2) a ∈ K ∗ → a  ∈ R ∗ +, 1 from K ∗ to R ∗ +, called the module of K. Its range (3) Mod(K) = {λ  ∈ R ∗ +; λ ∈ K ∗} is a closed subgroup of R ∗ +. The fields R, C and H (of quaternions) are the only ones with Mod(K) = R ∗ +, they are called Archimedian local fields. Let K be a non Archimedian local field, then (4) R = {x ∈ K; x  ≤ 1}, is the unique maximal compact subring of K and the quotient R/P of R by its unique maximal ideal is a finite field Fq (with q = p ℓ a prime power). One has, (5) Mod(K) = q Z ⊂ R ∗ +.
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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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.
On Dynamical Systems and Their Possible Significance for Arithmetic Geometry
, 1997
"... this paper is to first adress question b to some extent and then to investigate what properties dynamical systems must have in order that their dynamical cohomologies be isomorphic to the conjectured cohomology groups described in [D1], [D5]. In this program we are partly successful: In section two ..."
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Cited by 6 (0 self)
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this paper is to first adress question b to some extent and then to investigate what properties dynamical systems must have in order that their dynamical cohomologies be isomorphic to the conjectured cohomology groups described in [D1], [D5]. In this program we are partly successful: In section two we single out a class of dynamical systems that should contain all systems attached to schemes but is still much too vast. On the other hand from a dynamical systems point of view this class is very special. In particular the dynamical systems given by the geodesic flow on locally symmetric spaces do not belong to it which may account for the differences in the analytic behaviour of arithmetic and geometric zeta functions. After a study of this class in section three we pursue our strategy in section four and obtain topological information on the systems that should appear in a and some possibly useful hints as to their construction. Also our study shows that the present approach of realizing the conjectured cohomologies of [D1], [D5] as cohomologies of dynamical systems requires X 0 =ZZ to be in some sense ordinary if X
Inverse scattering, the coupling constant spectrum
 and the RH, Math. Phys., Analysis and Geometry
"... It is well known that the swave Jost function for a potential, λV, is an entire function of λ with an infinite number of zeros extending to infinity. For a repulsive V, and at zero energy, these zeros of the ”coupling constant”, λ, will all be real and negative, λn(0) < 0. By rescaling λ, such that ..."
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Cited by 2 (0 self)
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It is well known that the swave Jost function for a potential, λV, is an entire function of λ with an infinite number of zeros extending to infinity. For a repulsive V, and at zero energy, these zeros of the ”coupling constant”, λ, will all be real and negative, λn(0) < 0. By rescaling λ, such that λn < −1/4, and changing variables to s, with λ = s(s − 1), it follows that as a function of s the Jost function has only zeros on the line sn = 1 2 + iγn. Thus finding a repulsive V whose coupling constant spectrum coincides with the Riemann zeros will establish the Riemann hypothesis, but this will be a very difficult and unguided search. In this paper we make a significant enlargement of the class of potentials needed for a generalization of the above idea. We also make this new class amenable to construction via inverse scattering methods. We show that all one needs is a one parameter class of potentials, U(s;x), which are analytic in the strip, 0 ≤ Res ≤ 1, Ims> To, and in addition have an asymptotic expansion in powers of [s(s −1)] −1, i.e. U(s;x) = Vo(x) + gV1(x) + g 2 V2(x) +... + O(g N), with g = [s(s − 1)] −1. The potentials Vn(x) are real and summable. Under suitable conditions on the V ′ ns and the O(gN) term we show that the condition, ∫ ∞ o fo(x)  2V1(x)dx ̸ = 0, where fo is the zero energy and g = 0 Jost function for U, is sufficient to guarantee that the zeros gn are real and hence sn = 1 2 + iγn, for γn ≥ To. Starting with a judiciously chosen Jost function, M(s,k), which is constructed such that M(s,0) is Riemann’s ξ(s) function, we have used inverse scattering methods to actually construct a U(s;x) with the above properties. By necessity we had to generalize inverse methods to deal with complex potentials and a nonunitary Smatrix. This we have done at least for the special cases under consideration. For our specific example, ∫ ∞ o fo(x)  2V1(x)dx = 0, and hence we get no restriction on Imgn or Resn. The reasons for the vanishing of the above integral are given, and they give us hints on what one needs to proceed further. The problem of dealing with small but nonzero energies is also discussed. 1 I.
Use of Harmonic Inversion Techniques in Semiclassical Quantization and Analysis of Quantum Spectra
, 1999
"... Harmonic inversion is introduced as a powerful tool for both the analysis of quantum spectra and semiclassical periodic orbit quantization. The method allows to circumvent the uncertainty principle of the conventional Fourier transform and to extract dynamical information from quantum spectra which ..."
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Cited by 2 (1 self)
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Harmonic inversion is introduced as a powerful tool for both the analysis of quantum spectra and semiclassical periodic orbit quantization. The method allows to circumvent the uncertainty principle of the conventional Fourier transform and to extract dynamical information from quantum spectra which has been unattainable before, such as bifurcations of orbits, the uncovering of hidden ghost orbits in complex phase space, and the direct observation of symmetry breaking effects. The method also solves the fundamental convergence problems in semiclassical periodic orbit theories  for both the BerryTabor formula and Gutzwiller's trace formula  and can therefore be applied as a novel technique for periodic orbit quantization, i.e., to calculate semiclassical eigenenergies from a finite set of classical periodic orbits. The advantage of periodic orbit quantization by harmonic inversion is the universality and wide applicability of the method, which will be demonstrated in this work for v...
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.