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 random-matrix 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 Riemann-Siegel 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.

. In this article we describe what a cohomology theory related to zeta and L-functions 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: L-functions 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 ...

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In recent years there has been a growing interest in connections between the statistical properties of number theoretical L-functions and random matrix theory. We review the history of these connections, some of the major achievements, and a number of applications.

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 end-point 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 L-functions 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 ∗ +.

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

. In semiclassical theories for chaotic systems, such as Gutzwiller's periodic orbit theory, the energy eigenvalues and resonances are obtained as poles of a non-convergent series g(w) = # n A n exp(is n w). We present a general method for the analytic continuation of such a non-convergent series by harmonic inversion of the `time' signal, which is the Fourier transform of g(w). We demonstrate the general applicability and accuracy of the method on two different systems with completely different properties: the Riemann zeta function and the three-disk scattering system. The Riemann zeta function serves as a mathematical model for a bound system. We demonstrate that the method of harmonic inversion by filter-diagonalization yields several thousand zeros of the zeta function to about 12 digit precision as eigenvalues of small matrices. However, the method is not restricted to bound and ergodic systems, and does not require the knowledge of the mean staircase function, i.e. the Weyl t...

Introduction In this note we explain how the theory of the Riemann zeta function naturally leads to the investigation of a class of dynamical systems on foliated spaces. The hope is that finding the right dynamical system will be an important step towards a better understanding of i(s). The entire approach carries over to motivic L-series the most general kind of L-series coming from arithmetic geometry. This is important for various reasons but for simplicity we will mostly be concerned with i(s). In the first section we recall some arguments from [D2] in favour of a possible cohomological interpretation of the Riemann zeta function. In the second section following [D3], [D4] we single out a class of foliated dynamical systems whose leafwise reduced cohomology has many of the formal properties desired in section one. We close with a number of further remarks and suggestions. For other approaches to<F

Contents 1 Introduction 3 1.1 Motivation of semiclassical concepts . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Basic semiclassical theories . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.2 Convergence problems of the semiclassical trace formulae . . . . . . 5 1.2 Objective of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 High precision analysis of quantum spectra . . . . . . . . . . . . . . 7 1.2.2 Periodic orbit quantization . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 High precision analysis of quantum spectra 12 2.1 Circumventing the uncertainty principle . . . . . . . . . . . . . . . . . . . 13 2.2 Precision check of the periodic orbit theory . . . . . . . . . . . . . . . . . . 18 2.3 Ghost orbits and uniform semiclassical approximations . . . . . . . . . . . 22 2.3.1 The hyperbolic umbilic catastrophe . . . .

Chaos quantization conditions, which relate the eigenvalues of a Hermitian operator the Riemann operator with the non-trivial zeros of the Riemann zeta function are considered, and their geometrical interpretation is discussed. q 1999 Published by Elsevier Science B.V. All rights reserved.