By Pierre Emile Cartier, Pierre E. Cartier, Bernard Julia, Pierre Moussa, Pierre Vanhove

The relation among arithmetic and physics has an extended heritage, during which the function of quantity thought and of different extra summary elements of arithmetic has lately develop into extra prominent.More than ten years after a primary assembly in 1989 among quantity theorists and physicists on the Centre de body des Houches, a moment 2-week occasion inquisitive about the wider interface of quantity idea, geometry, and physics.This publication is the results of that intriguing assembly, and collects, in 2 volumes, prolonged types of the lecture classes, by way of shorter texts on distinctive subject matters, of eminent mathematicians and physicists. the current quantity has 3 components: Random matrices, Zeta features, Dynamical structures.

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**Extra info for Frontiers in Number Theory, Physics and Geometry: On Random Matrices, Zeta Functions and Dynamical Systems**

**Sample text**

2 πT −∞ 1 2π In the left hand side of the Selberg trace formula one obtains e−En T = 1 + n e−En T En >0 where we take into account that for any discrete group there is one zero eigenvalue corresponding to a constant eigenfunction. Therefore when T → ∞ the above sum tends to one T →∞ e−En T −→ 1 . n 30 Eugene Bogomolny One can easily check that in the right hand side of (29) the contribution of the smooth part of the density goes to zero at large T and the contribution of periodic orbits is important only for primitive periodic orbits with n = 1.

Separating the real and imaginary parts of the Schroedinger equation one gets two equations E − (∇S)2 − V (x) + 2 ∆A = 0 and 2∇S∇A + ∆SA = 0 . In the leading order in the ﬁrst equation reduces to the Hamilton-Jacobi equation for the classical action S(x, y) E = (∇S)2 + V (x) . It is well known that the solution of this equation can be obtained in the following way. Find the solution of the usual classical equations of motion ¨=− x ∂V ∂x with energy E which starts at a ﬁxed point y and ends at a point x.

This equation is the only function which has the following property. If z = x + iy belongs to this curve then z = az + b cz + d also belongs to it. The length of the periodic orbit is the distance along these geodesics between a point and its image. Let z as above be the result of transformation (26) then the distance between z and z is cosh lp = 1 + |z − z |2 . 2yy But y = y/|cz + d|2 and z− c(x + iy)2 − (d − a)(x + iy) − b −2cy + i(d − a + 2cx) az + b = =y . e. its coordinates obey (27)). Therefore 1 cosh lp = 1 + | − 2cy + i(d − a + 2acx)|2 2 1 1 = 1 + [4bc + (d − a)2 ] = (a + d)2 − 1 .