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1 Functional Equation The possibility of this continuation is connected with the fact that the Riemann zeta function satisﬁes the important functional equation ζ(s) = ϕ(s)ζ(1 − s) where (37) πs Γ (1 − s) . g. ). g. , 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 .
3 Trace Formulas for Integrable Dynamical Systems A f -dimensional system is called integrable if its classical Hamiltonian can be written as a function of action variables only H(I) = H(I1 , . . , If ) . In this representation the classical equations of motion take especially simple form ∂H ∂H ˙ = =0, ϕ =ω. I˙ = − ∂ϕ ∂I The semiclassical quantization consists of ﬁxing the values of the action variables µj Ij = (nj + ) 4 where nj are integers and µj are called the Maslov indices. In this approximation eigenvalues of energy of the system are a function of these integers E(n) = H (n1 + µ1 µf ), .
46], ). The simplest of them is the one-point correlation function or the mean ¯ level density, d(E), which is the probability density of ﬁnding a level in the ¯ interval (E, E + dE). When d(E) is known one can construct a new sequence of levels, en , called unfolded spectrum as follows en = En ¯ d(E)dE . This artiﬁcially constructed sequence has automatically unit local mean density which signiﬁes that the mean level density (provided it is a smooth function of E) plays a minor role in describing statistical properties of a spectrum at small intervals.
Algebraic inequalities by Vasile Cirtoaje