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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

##### MSC:
 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.)