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By Lucien Chambadal

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2 λ The variable u is connected with the distance by cosh d = 1 + u/2 and the function F (2(cosh d − 1)) has the form u0 = ∞ 1 F (2(cosh d − 1)) = √ 2 2π 2 d sin kr √ dr . cosh r − cosh d Introduce a variable τ connected with r as u is connected with d r , 2 cosh τ = 1 + It gives F (u) = 1 2π 2 ∞ dg (E) = f (w) = ∞ w √ sin kr(τ ) (τ − u)(τ 2 + 4τ ) u and where dr 1 =√ . 2 dτ τ + 4τ du u−w dτ ln λ20 √ f (u0 ) 2π 2 u0 ∞ u sin kr(τ ) (τ − u)(τ 2 + 4τ ) dτ . Changing the order of integration one obtains f (w) = ∞ w sin kr(τ ) √ dτ τ 2 + 4τ τ w du (u − w)(τ − u) .

One of important consequences of the above formula of analytical continuation is that it does not change under the substitution s → 1 − s. g. [32], Vol. 15) Γ (x)Γ (1 − x) = π , sin πx 1 Γ (2x) = 22x−1 π −1/2 Γ (x)Γ (x + ) 2 the last expression can be transformed to (38) which proves the functional equation (37). From the functional equation (37) is follows that ζ(s) has ’trivial’ zeros at negative even integers (except zero) s = −2, −4, . . which appear from sin(πs/2) in (38). All other non-trivial zeros, ζ(sn ) = 0, are situated in the socalled critical strip 0 < Re s < 1.

G. [16]) 2 p(s) = asβ e−bs where β is the same as above and constants a and b are determined from normalization conditions ∞ 0 p(s)ds = ∞ 0 sp(s)ds = 1 . These functions are presented at Fig. 9 together with the Poisson prediction for this quantity p(s) = e−s . 5 0 0 1 2 3 s Fig. 9. Nearest-neighbor distribution for the standard random matrix ensembles. Dotted line – the Poisson prediction Though random matrix ensembles were first introduced to describe spectral statistics of heavy nuclei later it was understood that the same conjectures can be applied also for simple dynamical systems and today’s standard accepted conjectures are the following • The energy levels of classically integrable systems on the scale of the mean level density behave as independent random variables and their distribution is close to the Poisson distribution [10].

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