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Abstract:
The paper under review discusses some of the fascinating appearances of the dilogarithm \\textLi\\sb 2(z) and polylogarithm \\textLi\\sb m(z) functions in various areas of mathematics. Special attention is paid to the Bloch-Wigner function D(z)=\\textIm(\\textLi\\sb 2(z))+\\arg (1-z)\\log \\vert z\\vert and its beautiful properties. The functional equations for \\textLi\\sb 2 take their cleanest form when D is considered as a function of the cross-ratio of four complex numbers. There are also analogues D\\sb m of D built up out of \\textLi\\sb m. These functions are related with higher weight Green's functions for the hyperbolic Laplacian on H/Γ where H is the upper half-plane and Γ a congruence subgroup of \\textSL\\sb 2(\\Bbb Z). The function D also comes up in connection with measurements of volumes in the upper half-space model \\Bbb C×]0,∞ [ of hyperbolic geometry: D(z) is equal to the hyperbolic volume of an ideal tetrahedron with vertices ∞, 0, 1, z. This in turn has interesting consequences for the volume spectrum of hyperbolic 3-manifolds. In the last section it is shown that certain special values of Dedekind's zeta function ζ\\sb F of an algebraic number field F may be expressed in terms of the function D, and the role of the Bloch group is explained. Moreover, the author formulates a general conjecture on a relation of ζ\\sb F(m) with special values of the Bloch-Wigner-Ramakrishnan function D\\sb m(z) with suitable arguments z\\in F. \\par This attractive survey paper will be of interest to a broad readership.
