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Polylogarithmic functions, polyzeta numbers and pro-unipotent groups. (Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents.) (French) Zbl 1085.11042
Bourbaki seminar. Volume 2000/2001. Exposés 880-893. Paris: Société Mathématique de France (ISBN 2-85629-130-9/pbk). Astérisque 282, 137-173, Exp. No. 885 (2002).
This text is a lively survey on polyzetas, polylogarithms, and their values (state of the art in 2000–2001). The author goes from the classical zeta values and their computations to polyzetas and polylogarithms, from Zagier conjecture to shuffle products, from noncommutative generating power series to colored polyzetas. This makes this paper and the commented bibliography a useful introduction to this appealing subject.
Please note that two papers by the same author can be added to Reference [B3] [J. Ecalle, J. Théor. Nombres Bordx. 15, No. 2, 411–478 (2003; Zbl 1094.11032) and Ann. Fac. Sci. Toulouse, VI. Sér., Math. 13, No. 4, 683–708 (2004; Zbl 1161.11380)], and to [B8] [M. E. Hoffman and Y. Ohno, J. Algebra 262, No. 2, 332–347 (2003; Zbl 1139.11322)] and to [B18] [J. M. Borwein, D. M. Bradley and D. J. Broadhurst, Electron. J. Comb. 4, No. 2, Research paper R5, 19 p (1997); printed version J. Comb. 4, No. 2, 31–49 (1997; Zbl 0884.40004)].
Also note that [C5] can be completed by a published paper [M. Bigotte, G. Jacob, N. E. Oussous and M. Petitot, Theor. Comput. Sci. 273, No. 1–2, 271–282 (2002; Zbl 1014.68126)], and that [C7] appeared with a slightly different title [Hoang Ngoc Minh, M. Petito and J. van der Hoeven, Discrete Math. 225, No. 1–3, 217–230 (2000; Zbl 0965.68129)]. Finally, note that [D6] appeared in [U. Müller and C. Schubert, Int. J. Math. Math. Sci. 31, No. 3, 127–148 (2002; Zbl 1085.05016)].
For the entire collection see [Zbl 1007.00024].

##### MSC:
 11M32 Multiple Dirichlet series and zeta functions and multizeta values 11-02 Research exposition (monographs, survey articles) pertaining to number theory 33B30 Higher logarithm functions 11G55 Polylogarithms and relations with $$K$$-theory 19F27 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects) 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11R42 Zeta functions and $$L$$-functions of number fields 19D45 Higher symbols, Milnor $$K$$-theory
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