The solutions to this equation require the solving of a geodesic flow in an infinite dimensional manifold. These solutions enable one approach to a formula generating the prime numbers. Broughan and A. Attention is given to the characterization of the flow lines in the neighborhood of the first zeros on the critical line. All of these zeros are foci. The majority are sources, but in a small proportion of exceptional cases, the zero is a sink.

To produce these portraits, the zeta function was evaluated numerically to 12 decimal places, in the region of interest using the Chebyshev method and using Mathematica. The phase diagrams suggest new analytic properties of zeta, a number of which are proved and a number of which are given in the form of conjectures. Broughan's research page includes further connections between dynamical systems and the zeta and xi functions and more phase portraits of the zeros of these functions L.

## CiteSeerX — VALUATIONS AND HYPERBOLICITY IN DYNAMICS

Bunimovich and C. Dettmann, "Open circular billiards and the Riemann hypothesis" , Phys. The corresponding quantities are expressed as sums over zeroes of the Riemann zeta function.

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Thus we demonstrate a direct connection between recent experiments and major unsolved problems in number theory. The exact formulas for escape are obtained which involve the Riemann zeta function and Dirichlet L-functions. It is shown that the problem of finding the exact asymptotics in the small hole limit for escape in some of these billiards is equivalent to the Riemann hypothesis.

The results are conditional on a number of conjectures, and are corroborated by numerical simulations in up to ten dimensions. Petkov, "Analytic singularities of the dynamical zeta function" , Nonlinearity 12 Verjovsky, "Arithmetic, geometry and dynamics in the unit tangent bundle of the modular orbifold" , Dyamical Systems. Proceedings of the 3rd international school of dynamical systems, Santiago de Chile, R.

Bamon, et. Longman Scientific and Tehcnical Pitman Res. Notes Math. This of course does not disprove the Riemann hypothesis. The proof is based on reducing the problem to a lattice point counting problem. The impossibility of improving the error is related to an analogous situation in the circle problem related to the slow decay of Fourier transforms of characteristic functions. It mainly refers to the symbolic dynamics theory. By this method, we could connect the prime system with familiar 'Logistic Mapping'.

An interesting discovery is that the pattern of primes could be depicted by a series of orbits of this mapping. Some heuristic proofs for open problems like twin primes are obtained through this relation. This research gives a new viewpoint for the distribution of prime numbers. We establish a link between transfer probabilities of order n and some sets of sequences of the symbolic dynamic.

This leads to a way of computing these coefficients using Monte Carlo method. We propose an algorithm which leads to a density indicating a good experimental fit with a typical orbit. Garcia-Perez, M. Angeles Serrano and M. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as building blocks remains elusive. Here, we propose a new approach to decoding the architecture of natural numbers based on complex networks and stochastic processes theory. We introduce a parameter-free non-Markovian dynamical model that naturally generates random primes and their relation with composite numbers with remarkable accuracy.

Regarding composites, the model helps us to derive the prime factors counting function, giving the probability of distinct prime factors for any integer. Probabilistic models like ours can help not only to conjecture but also to prove results about primes and the complex architecture of natural numbers. Bonanno and M. Mega, "Toward a dynamical model for prime numbers" , Chaos, Solitons and Fractals 20 [abstract:] "We show one possible dynamical approach to the study of the distribution of prime numbers.

Our approach is based on two complexity methods, the Computable Information Content and the Entropy Information Gain, looking for analogies between the prime numbers and intermittency. For this to work, the domain of T z [0,1] must be partitioned as [0,0. The authors attempt to exploit the resemblance to the approximating function in the Prime Number Theorem , and in some sense model the distribution of primes in dynamical terms, i. Certain refinements of this are then explored.

Gallas, "On the origin of periodicity in dynamical systems" , Physica A: Statistical Mechanics and its Applications [abstract:] "We prove a theorem establishing a direct link between macroscopically observed periodic motions and certain subsets of intrinsically discrete orbits which are selected naturally by the dynamics from the skeleton of unstable periodic orbits UPOs underlying classical and quantum dynamics.

As a simple illustration, an infinite set of UPOs of the quadratic logistic map is used to build ab initio the familiar trigonometric and hyperbolic functions and to show that they are just the first members of an infinite hierarchy of functions supported by the UPOs. Although all microscopic periodicities of the skeleton involve integer discrete periods only, the macroscopic functions resulting from them have real non-discrete periods proportional to very complicate non-integer numbers, e. Itzykson, et. Springer, "We shall start by briefly summarizing the results of the "local" renormalization theory for transitions from quasiperiodicity to chaos.

In experimental tests of this theory one adjusts the external frequency to make the frequency ratio as far as possible from being mode-locked. The choice of the golden mean is dictated by number theory: the golden mean is the irrational number for which it is hardest to give good rational approximants. As experimental measurements have limited accuracy, physicists usually do not expect that such number-theoretic subtleties as how irrational a number is should be of any physical interest.

However, in the dynamical systems theory to chaos the starting point is the enumeration of asymptotic motions of a dynamical system, and through this enumeration number theory enters and comes to play a central role. Number theory comes in full strength in the 'global' theory of circle maps, the study of universal properties of the entire irrational winding set - the main topic of these lectures.

The universality or even existence of this dimension has not yet been rigorously extablished. We shall offer here a rather pretty explanation of this universality or even existence of this dimension has not yet been rigorously established. We shall offer here a rather pretty explanation of this universality in form of the explicit formula 39 which expresses this Hausdorff dimension as an average over the Shenker universal scaling numbers.

The renormalization theory of critical circle maps demands at present rather tedious numerical computations, and our intuition is much facilitated by approximating circle maps by number-theoretic models. The model that we shall use here to illustrate the basic concepts might at first glance appear trivial, but we find it very instructive, as much that is obscured by numerical work required by the critical maps is here readily number-theoretically accessible. Indicative of the depth of mathematics lurking behind physicists' conjectures is that fact that the properties that one would like to establish about the renormalization theory of critical circle maps might turn out to be related to number-theoretic abysses such as the Riemann conjecture, already in the context of the 'trivial' models.

We prove that such a system is linearly repetitive if and only if it is minimal. Based on this characterization we extend various results from primitive substitutions to minimal substitutions. Everest, Y. Puri and T. Ward, "Integer sequences counting periodic points" [Abstract:] "An existing dialogue between number theory and dynamical systems is advanced.

A combinatorial device gives necessary and sufficient conditions for a sequence of non-negative integers to count the periodic points in a dynamical system. This is applied to study linear recurrence sequences which count periodic points. Instances where the p -parts of an integer sequence themselves count periodic points are studied.

The Mersenne sequence provides one example, and the denominators of the Bernoulli numbers provide another. The methods give a dynamical interpretation of many classical congruences such as Euler-Fermat for matrices, and suggest the same for the classical Kummer congruences satisfied by the Bernoulli numbers. Beukers, J. Sanders and J. Wang, "On integrability of systems of evolution equations" , Journal of Differential Equations [abstract:] "We prove the conjecture We list the nine exceptional cases when the systems do have infinitively many symmetries.

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For such systems, we give the recursive operators to generate their symmetries. We treat both the commutative and the noncommutative or quantum cases.

This is the first example of a class of equations where such a classification has been possible. Percival and F. Vivaldi, "Arithmetical properties of strongly chaotic motions", Physica D 25 We transform the dynamics of the periodic orbits of these maps into modular arithmetic in suitable domains of quadratic integers, classify all periodic orbits, and show how to determine their periods and initial conditions. The methods are based on ideal theory in quadratic fields, which is reviewed.

It is shown that the structure of orbits rests upon some basic arithmetical notions, such as unique factorization into prime ideals. Rudnick , "Value distribution for eigenfunctions of desymmetrized quantum maps" "We study the value distribution and extreme values of eigenfunctions for the "quantized cat map". This is the quantization of a hyperbolic linear map of the torus.

## Prizes of the European Mathematical Society

In a previous paper it was observed that there are quantum symmetries of the quantum map - a commutative group of unitary operators which commute with the map, which we called "Hecke operators". The eigenspaces of the quantum map thus admit an orthonormal basis consisting of eigenfunctions of all the Hecke operators, which we call "Hecke eigenfunctions".

In this note we investigate suprema and value distribution of the Hecke eigenfunctions. For prime values of the inverse Planck constant N for which the map is diagonalizable modulo N the "split primes" for the map , we show that the Hecke eigenfunctions are uniformly bounded and their absolute values amplitudes are either constant or have a semi-circle value distribution as N tends to infinity.

Moreover in the latter case different eigenfunctions become statistically independent.

### VALUATIONS AND HYPERBOLICITY IN DYNAMICS

We obtain these results via the Riemann hypothesis for curves over a finite field Weil's theorem and recent results of N. Katz on exponential sums. For general N we obtain a nontrivial bound on the supremum norm of these Hecke eigenfunctions.