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Structural properties of polylogarithms. (English) Zbl 0745.33009
Mathematical Surveys and Monographs. 37. Providence, RI: American Mathematical Society (AMS). xviii, 412 p. (1991).
This marvelous book celebrates the re-emergence of the polylogarithm functions $$Li_ k(z)=\sum_{n=1}^ \infty z^ n/n^ k$$. The editor, Leonard Lewin, records a fascination for these functions over 60 years and has assembled a wealth of history and a remarkable zoo of peculiar numerical identities, such as Watson’s example $Li_ 2(\alpha)-Li_ 2(\alpha^ 2)=\pi^ 2/42+\log^ 2\alpha, \alpha={1 \over 2}sec {{2\pi} \over 7}.$ The first part of the book recounts the work of Lewin and Abouzahra in introducing a structure amongst these identities and relating them to the functional equations satisfied by the polylogarithm. It has been a mystery how so many identities could stem from the functional equations, as it seems they should. Lewin’s systematic studies and a great deal of computer algebra have recently yielded new categories of complicated functional equations. Nevertheless, many identities remain numerically sound, but unproven. In another chapter, Ray approaches the identities by specialising a different class of functional identities with considerable, but still only partial success. Wechsung draws on connections with hyperbolic geometry and automorphic functions and analyses the singularities of the polylogarithms. He shows how to construct functional equations of the classical Kummer type involving 2 independent variables for $$Li_ 2,Li_ 3,Li_ 4$$ and $$Li_ 5$$ and proves that there are no such functional equations for $$Li_ n$$ for $$n\geq 6$$. A more topological approach is given by Wojtkowiak. Further connections with volumes of hyperbolic polytopes are made by Kellerhals. Zagier, building on work of Bloch and Browkin, has given a grand conjectural framework relating special values of polylogarithms to regulators in algebraic number fields. All this and the underlying $$K$$- theory is taken up by Zagier, Browkin, Bloch and Hain and MacPherson, Beilinson and Deligne have begun to account for these conjectures, but many remain unresolved. One remarkable result from these insights is the new 2 variable functional equation for $$Li_ 6$$ discovered by Gangl. It is the first such example beyond the classical equations at order 5 and has 87 terms.
This is a remarkable book on a remarkable function, brimming with the enthusiasm of new discoveries and ideas for further research.
Contents: 1. L. Lewin, The evolution of the ladder concept (1–10); 2. L. Lewin, Dilogarithmic ladders (11–25); 3. M. Abouzahra and L. Lewin, Polylogarithmic ladders (27–47); 4. M. Abouzahra and L. Lewin, Ladders in the trans-Kummer region (49–68); 5. M. Abouzahra and L. Lewin, Supernumary ladders (69–96); 6. L. Lewin, Functional equations and ladders (97–121); 7. G. A. Ray, Multivariable polylogarithm identities (123–169); 8. G. Wechsung, Functional equations of hyperlogarithms (171–184); 9. G. Wechsung, Kummer-type functional equations of polylogarithms (185–203); 10. Zdzisław Wojtkowiak, The basic structure of polylogarithmic functional equations (205–231); 11. J. Browkin, $$K$$-theory, cyclotomic equations, and Clausen’s function (233–273); 12. Spencer Bloch, Function theory of polylogarithms (275–285); 13. J. H. Loxton, Partition identities and the dilogarithm (287–299); 14. Ruth Kellerhals, The dilogarithm and volumes of hyperbolic polytopes (301–336); 15. Richard M. Hain and Robert MacPherson, Introduction to higher logarithms (337–353); 16. L. Lewin, Some miscellaneous results (355–375); Appendix A. Don Zagier, Special values and functional equations of polylogarithms (377–400); Appendix B. R. MacPherson and H. Sah, Summary of the Informal Polylogarithm Workshop, November 17–18, 1990, MIT, Cambridge, Massachusetts (401–404).

##### MSC:
 33E20 Other functions defined by series and integrals 11R42 Zeta functions and $$L$$-functions of number fields 33-02 Research exposition (monographs, survey articles) pertaining to special functions 33-06 Proceedings, conferences, collections, etc. pertaining to special functions 00B15 Collections of articles of miscellaneous specific interest 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11R70 $$K$$-theory of global fields 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 19F27 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects) 33B99 Elementary classical functions