[...]... models, hoping that for certain questions a problem will be more tractable than another Second, one considers, instead of exact quantities, the calculation of their smoothed values In many cases such coarse graining appears naturally in experimental settings and, usually, it is more easy to treat Third, one tries to understand statistical properties of quantum quantities by organizing them in suitable... rectangular billiard with periodic boundary conditions calculated by taking into account 250 different periodic orbits Dotted lines indicate the position of exact energy levels 2 Billiards on Constant Negative Curvature Surfaces The crucial point in the second method of derivation of the trace formula for the rectangular billiard with periodic boundary conditions was a representation of the exact Green function... system is derived The derivation is presented in a manner which permits to generalize it to the Selberg case of constant negative curvature surfaces generated by discrete groups which is considered in details in Sect 2 In Sects 3 and 4 the derivations of the trace formula for, respectively, classically integrable and chaotic systems are presented In Sect 5 it is demonstrated that the density of Riemann... this Chapter is devoted to a heuristic derivation of the ‘exact’ twopoint correlation function for the Riemann zeros The derivation is based on the Hardy–Littlewood conjecture about the distribution of prime pairs which is also reviewed The resulting formula agrees very well with numerical calculations of Odlyzko In Chapter III a special class of dynamical systems is considered, namely, hyperbolic... the calculation by Keating and Snaith of moments of the Riemann zeta function using random matrix theory [43]) The topics of these lectures were chosen specially to emphasize the interplay between physics and mathematics which is typical in quantum chaos In Chapter I different types of trace formulas are discussed The main attention is given to the derivation of the Selberg trace formula which relates... exact Green function one has to note that (i) it obeys (18) because each term in the sum obeys it, (ii) it obeys boundary conditions (19) by construction (provided the sum converges), and (iii) inside the initial rectangle only identity term can produce a δ-function contribution required in (18) because all other terms will give δ-functions outside the rectangle The next steps are straightforward The... to number theory, in particular, to the zeros of the Riemann zeta function Though a hypothetical quantum-like system whose eigenvalues coincide with the imaginary part of Riemann zeta function zeros has not (yet!) been found, the Riemann zeta function is, in many aspects, similar to dynamical zeta functions and the investigation of such relations already mutually enriched both quantum chaos and number. .. 2 The Poincar´ model of constant negative curvature space Solid line indicates e the geodesic passing through points A and B y = (x2 , y2 ) is defined as the length of the geodesic connecting these points Explicitly Quantum and Arithmetical Chaos cosh d(x, y) = 1 + (x1 − x2 )2 + (y1 − y2 )2 |z1 − z2 |2 =1+ 2y1 y2 2Im z1 Im z2 17 (22) where in the last equation one combined coordinates (x, y) into a complex... = 1 and the corresponding group is discrete But if α is an irrational number, the images of any point cover the whole circle uniformly and the group is not discrete Modular Group Mathematical fact: in the upper half plane there exists an in nite number of discrete groups (see e.g [42]) As an example let us consider the group of 2 × 2 integer matrices with unit determinant mn k l , m, n, k, l are integers... that points z and z where z = az + b cz + d (26) are identified i. e they are considered as one point The classical motion on the resulting surface is the motion (with unit velocity) on geodesics (semi-circles perpendicular to the real axis) inside the fundamental domain but when a trajectory hits a boundary it reappears from the opposite side as prescribed by boundary identifications For each hyperbolic . Frank Institute and Enrico Fermi Institute of the University of Chicago, 5640 S.Ellis Avenue, Chicago, IL 60637, USA Landau Institute for Theoretical Physics,. theory and families of arithmetic L-functions (mostly in characteristics zero), that is Dirichlet series satisfying functional equations similar to those