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 booklet provides pedagogical contributions on chosen issues concerning quantity concept, Theoretical Physics and Geometry. The elements are composed of lengthy self-contained pedagogical lectures via shorter contributions on particular matters prepared by way of subject matter. so much classes and brief contributions move as much as the hot advancements within the fields; a few of them persist with their author?s unique viewpoints. There are contributions on Random Matrix conception, Quantum Chaos, Non-commutative Geometry, Zeta features, and Dynamical platforms. The chapters of this ebook are prolonged types of lectures given at a gathering entitled quantity concept, Physics and Geometry, held at Les Houches in March 2003, which collected mathematicians and physicists.

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G. 1, Sect. 3) (1 − u2 ) d2 G dG + l(l + 1)G = 0 − 2u du2 du where E= 1 + k 2 = −l(l + 1) 4 and 1 l = − − ik . 2 As for the plane case the required solution of the above equation should grow as eikd when d → ∞ and should behave like ln d/2π when d → 0. 1, Sect. 3 it follows that (0) GE (x, x ) = − 1 Q 1 (cosh d(x, x )) . 2π − 2 −ik Here Q− 12 −ik (cosh d) is the Legendre function of the second kind with the integral representation [32], Vol. 4) 1 Q− 12 −ik (cosh d) = √ 2 ∞ d √ eikr dr cosh r − cosh d and the following asymptotics d→0 Q− 12 −ik (cosh d) −→ − log d and Quantum and Arithmetical Chaos d→∞ Q− 12 −ik (cosh d) −→ 23 π ei(kd−π/4) .

But pi = − ∂S , ∂yi pf = ∂S . ∂yf Therefore δpi = − ∂2S ∂2S ∂2S ∂2S δy − δy , δp = δy + δyf . i f f i ∂yi2 ∂yi ∂yf ∂yi ∂yf ∂yf2 From comparison of these two expression one obtains the expressions of the second derivatives of the action through monodromy matrix elements y periodic orbit classical orbit Fig. 7. A periodic orbit and a closed classical orbit in its vicinity Quantum and Arithmetical Chaos 41 1 ∂2S m11 ∂2S m22 ∂2S =− , = , = . 2 ∂yi ∂yf m12 ∂yi m12 ∂yf2 m12 Substituting these expressions to the contribution to the trace formula from one periodic orbit one gets (in two dimensions) (E) = d(osc) p 1 i(2πi )3/2 i m11 + m22 − 2 2 dx |m12 |−1/2 exp( Sp + i y )dy 2 m12 k(x) where x and y are respectively coordinates parallel and perpendicular to the trajectory.

32], Vol. 1, Sect. 1). Therefore if Re s > 1 ∞ Γ (s/2)ζ(s) = xs/2−1 Ψ (x)dx π s/2 0 where Ψ (x) is given by the following series ∞ Ψ (x) = e−πn 2 x . n=1 Using the Poisson summation formula (5) one obtains ∞ e−πn 2 n=−∞ x ∞ 2 1 =√ e−πn /x x n=−∞ which leads to the identity 1 2Ψ (x) + 1 = √ x 1 2Ψ ( ) + 1 x . Hence 1 ξ(s) ≡ π −s/2 Γ ( s)ζ(s) = 2 1 = 0 s/2 x 1 0 xs/2 Ψ (x)dx + 1 1 1 1 √ Ψ( ) + √ − x x 2 x 2 ∞ 1 xs/2 Ψ (x)dx = ∞ dx + 1 xs/2 Ψ (x)dx 1 ∞ 1 1 1 − + = xs/2−3/2 Ψ ( )dx + xs/2 Ψ (x)dx = s−1 s x 0 1 ∞ 1 = + x−s/2−1/2 + xs/2−1 Ψ (x)dx .

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