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Multiple zeta values: an introduction. (Valeurs zêta multiples. Une introduction.) (French) Zbl 0976.11037
For positive integers $$s_1,\ldots,s_k$$ with $$s_1\geq 2$$, the multiple zeta values are $\zeta(s_1,\ldots,s_k)=\sum_{n_1>\ldots>n_k\geq 1} n_1^{-s_1}\cdots n_k^{-s_k}.$ The product of two zeta values is a sum of zeta values which is obtained by multiplying out the two series (e.g. $$\zeta(s)\zeta(s')=\zeta(s,s')+\zeta(s',s)+\zeta(s+s')$$). Another linear relation arises from the iterated integral representation for $$\zeta({\mathbf s})$$. Write $$x_{\mathbf s}=x_{\varepsilon_1}\cdots x_{\varepsilon_p}$$ where $${\mathbf s}=(s_1,\ldots,s_k)$$, $$p=s_1+\cdots+s_k$$ and $$\varepsilon=0$$ or 1. Set $$\omega_0(t)=dt/t$$ and $$\omega_1(t)=dt/(1-t)$$ and let $$\Delta_p$$ denote the simplex $$\{{\mathbf t}:1>t_1>\cdots>t_p>0\}$$ in $$\mathbb{R}^p$$. Then $\zeta({\mathbf s})=\int_{\Delta_p}\omega_{\varepsilon_1}(t_1)\cdots \omega_{\varepsilon_p}(t_p).$ Writing this as an iterated integral leads to linear relations such as $$\zeta(2)\zeta(3)=\zeta(2,3)+3\zeta(3,2)+6\zeta(4,1)$$. The main conjecture is that these two types of relations are sufficient to describe all algebraic relations between these numbers. The paper goes on to explore some algebraic implications of these ideas. There are intriguing connections with the theories of polylogarithms of A. B. Goncharov [Math. Res. Lett. 5, 497-516 (1998; Zbl 0961.11040)] and D. B. Zagier [Proc. First European Congress of Mathematicians, Vol. 2, Prog. Math. 120, 497-512 (1994; Zbl 0822.11001)].

##### MSC:
 11M41 Other Dirichlet series and zeta functions 33B30 Higher logarithm functions
##### Keywords:
multiple zeta values; algebraic relations; polylogarithms
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##### References:
 [1] Euler, L., Meditationes circa singulare serierum genus. In: Leonhardi Euleri Opera Omnia, Series Prima XV, Commentationes Analyticae Vol. 2, 217-267; Novi Comm. Acad. Sci. Petropol.20 (1775), 140-186. [2] Drinfeld, V.G., On quasitriangular Quasi-Hopf algebras and a group closely connected with Gal(/Q). Leningrad Math. J.24 (1991), 829-860. · Zbl 0728.16021 [3] Zagier, D., Values of zeta functions and their applications. Proc. First European Congress of Mathematics, Vol. 2, Birkhäuser, Boston (1994), 497-512. · Zbl 0822.11001 [4] Goncharov, A.B., Polylogarithms in arithmetic and geometry. roc. ICM-94 Zürich, Vol. 1, Birkhäuser, Boston (1995), 374-387. · Zbl 0849.11087 [5] Bar-Natan, D., On associators and the Grothendieck-Teichmuller Group I. In http://xxx. lanl. gov/abs/q-alg/9606021 and http://www.ma.huj i. ac. il/ drorbn · Zbl 0974.16028 [6] Hoffman, M.E., The Algebra of Multiple Harmonic Series. J. Algebra194 (1997), 477-495. · Zbl 0881.11067 [7] Hoang Ngoc, Minh, Petitot, M., Van Der Hoeven, J., Shuffle algebra and polylogarithms. Proc. of FPSAC’98, 10-th international Conference on Formal Power Series and Algebraic Combinatorics, June 98, Toronto. · Zbl 0965.68129 [8] Borwein, J.M., Bradley, D.M., Broadhurst, D.J., Lisonk, P., Combinatorial Aspects of Multiple Zeta Values. The Electronic Journal of Combinatorics5 (1998), #R38. · Zbl 0904.05012 [9] Goncharov, A.B., Multiple polylogarithms, cyclotomy and modular complexes. Math. Research Letter5 (1998), 497-516. · Zbl 0961.11040 [10] Hoang Ngoc, Minh, Petitot, M., Van Der Hoeven, J., L’algèbre des polylogarithmes par les séries génératrices. Proc. of FPSAC’99, 11-th international Conference on Formal Power Series and Algebraic Combinatorics, June 99, Barcelona. [11] Kontsevich, M., Periods. Journée annuelle de la Société Mathématique de France1999, 28-39. · Zbl 1058.11501 [12] Müller, U., Schubert, C., A Quantum Field Theoretical Representation of Euler-Zagier Sums. http: //xxx. lanl. gov/abs/math/9908067 · Zbl 1085.05016 [13] Hoang Ngoc, Minh, Petitot, M., Lyndon words, polylogarithms and the Riemann ζ function. Formal power series and algebraic combinatorics (Vienna, 1997). Discrete Math.217 (2000), 273-292. · Zbl 0959.68144 [14] J, Écalle, La libre génération des multizêtas et leur décomposition canonico-explicite en irréductibles. Manuscrit, 1999. [15] Hoang Ngoc, Minh, Petitot, M., Contribution à l’étude des MZV. Manuscrit, 1999. [16] Hoang Ngoc, Minh, Jacob, G., Petitot, M., Oussous, N.E., Aspects combinatoires des polylogarithmes et des sommes d’Euler-Zagier. Sém. Lothar. Combin.43 (1999), Art. B43e, 29 pp. (electronic). · Zbl 0964.33003
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