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Let Γ\H 3 be a finite-volume quotient of the upper-half space, where Γ ⊂ SL(2, C) is a discrete subgroup. To a finite dimensional unitary representation χ of Γ one associates the Selberg zeta function Z(s; Γ; χ). In this paper we prove the Artin formalism for the Selberg zeta function. Namely, if Γ is a finite index group extension of Γ in SL(2, C), and π = Ind Γ Γ χ is the induced representation, then Z(s; Γ; χ) = Z(s; Γ; π). In the second part of the paper we prove by a direct method the analogous identity for the scattering function, namely φ(s; Γ; χ) = φ(s; Γ; π), for an appropriate normalization of the Eisenstein series.

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In the present paper we deduce explicit formulas for the probability laws of the quotients Xt/Rt and mt/Rt, where Xt is the standard Brownian motion and mt, Mt, Rt are its running minimum, maximum and range, respectively. The computation makes use of standard techniques from analytic number theory and the theory of the Hurwitz zeta function.