ARI/GARI, dimorphy and multiple zeta arithmetic: a first evaluation. (ARI/GARI, la dimorphie et l’arithmétique des multizêtas: un premier bilan.) (French) Zbl 1094.11032

Ecalle, Jean

J. Théor. Nombres Bordx. 15, No. 2, 411-478 (2003).

This is an expository article on multiple zeta values (MZV): \[ \zeta(s_1,\dots,s_r;e_1,\dots,e_r)= \sum_{0<n_r<\cdots<n_1}\frac{e_r^{n_r}}{n_r^{s_r}} \cdots\frac{e_1^{n_1}}{n_1^{s_1}}, \] where \(e_j= \exp(2\pi i \varepsilon_j)\) is a root of unity. These MZVs have a remarkable property called ‘dimorphy’, i.e. they satisfy two kinds of ‘quadratic relations’. To study these relations, especially from its formal (symbolic) aspects, the author introduces a new structure: the Lie algebra ARI and its group GARI.

Reviewer: Makoto Ohtsuki (Kodaira)

MSC:

11M41	Other Dirichlet series and zeta functions
17B99	Lie algebras and Lie superalgebras
33B15	Gamma, beta and polygamma functions

Keywords:

multiple zeta values; dimorphy; Lie algebra

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