Frontiers in number theory, physics, and geometry I by Pierre Emile Cartier, Pierre E. Cartier, Bernard Julia,

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

This e-book offers pedagogical contributions on chosen subject matters bearing on quantity thought, Theoretical Physics and Geometry. The components are composed of lengthy self-contained pedagogical lectures through shorter contributions on particular topics equipped through topic. so much classes and brief contributions move as much as the new advancements within the fields; a few of them stick with their author?s unique viewpoints. There are contributions on Random Matrix concept, Quantum Chaos, Non-commutative Geometry, Zeta services, and Dynamical platforms. The chapters of this e-book are prolonged models of lectures given at a gathering entitled quantity conception, Physics and Geometry, held at Les Houches in March 2003, which accrued mathematicians and physicists.

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This is the famous Selberg trace formula. It connects eigenvalues of the Laplace–Beltrami operator for functions automorphic with respect to a discrete group having only hyperbolic elements with classical periodic orbits. 9 Density of Periodic Orbits To find the density of periodic orbits for a discrete group let us choose the test function h(r) in (29) as h(r) = e−(r 2 +1/4)T ≡ e−ET with a parameter T > 0. Its Fourier transforms is g(u) = ∞ 2 e−T /4 h(k)e−iku dk = √ e−u /4T . 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.

Assume that g is in the diagonal form. Then g(z) = λ2 z and cosh d(z, g(z)) = 1 + (λ2 − 1)2 (x2 + y 2 ) . 2λ2 y 2 Because λ0 is real the transformation z = λ20 z gives y = λ20 y and the funda2 mental domain of Sg = λ2m 0 z has the form of a horizontal strip 1 < y < λ0 indicated in Fig. 4. Now dg (E) = λ20 ∞ dx −∞ 1 (λ2 − 1)2 (x2 + y 2 ) λ2 y 2 F dy . y2 Introducing a new variable ξ = xy one gets dg (E) = λ20 1 = ln λ20 After the substitution dy y ∞ ∞ −∞ −∞ F F (1 + ξ 2 ) (1 + ξ 2 ) (λ2 − 1)2 λ2 (λ2 − 1)2 λ2 dξ dξ .

G. [55]) x . N (p < x) = ln x As ln p ≡ Tp this expression has the form similar to number of periodic orbits of chaotic systems with h = 1 N (Tp < T ) = eT . T Due to these similarities number-theoretical zeta functions play the role of simple (but by far non-trivial) models of quantum chaos. Notice that the overall signs of the oscillating part of trace formulas for the Riemann zeta function and dynamical systems are different. According to Connes [30] it may be interpreted as Riemann zeros belong not to a spectrum of a certain self-adjoint operator but to an ’absorption’ spectrum.

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