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The remarkable dilogarithm. (English) Zbl 0669.33001
The paper under review discusses some of the fascinating appearances of the dilogarithm \(\text{Li}_ 2(z)\) and polylogarithm \(\text{Li}_ m(z)\) functions in various areas of mathematics. Special attention is paid to the Bloch-Wigner function \[ D(z)=\text{Im}(\text{Li}_ 2(z))+\arg (1-z)\log | z| \] and its beautiful properties. The functional equations for \(\text{Li}_ 2\) take their cleanest form when \(D\) is considered as a function of the cross-ratio of four complex numbers. There are also analogues \(D_ m\) of \(D\) built up out of \(\text{Li}_ m\). These functions are related with higher weight Green’s functions for the hyperbolic Laplacian on \(H/\Gamma\) where \(H\) is the upper half-plane and \(\Gamma\) a congruence subgroup of \(\text{SL}_ 2({\mathbb Z})\). The function \(D\) also comes up in connection with measurements of volumes in the upper half-space model \({\mathbb C}\times]0,\infty [\) of hyperbolic geometry: \(D(z)\) is equal to the hyperbolic volume of an ideal tetrahedron with vertices \(\infty, 0, 1, z\). 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 \(\zeta_ F\) 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 \(\zeta_ F(m)\) with special values of the Bloch-Wigner-Ramakrishnan function \(D_ m(z)\) with suitable arguments \(z\in F\).
This attractive survey paper will be of interest to a broad readership.
Reviewer: J.Elstrodt

33-02 Research exposition (monographs, survey articles) pertaining to special functions
33B15 Gamma, beta and polygamma functions
11G55 Polylogarithms and relations with \(K\)-theory
11R42 Zeta functions and \(L\)-functions of number fields
11F11 Holomorphic modular forms of integral weight
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)