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Linear forms in multiple zeta values and multiple integrals. (Formes linéaires en polyzêtas et intégrales multiples.) (French) Zbl 1017.11048
R. Apéry’s proof of the irrationality of $$\zeta(3)$$ [Astérisque 61, 11-13 (1979; Zbl 0401.10049)] has given fresh impetus to explorations of linear relations between the numbers $$\zeta(n)$$. The author compares three families of $$n$$-dimensional integrals, of which one is $L({\mathbf p})=L(a_1,\ldots,a_n,b_1,\ldots,b_n,c_2,\ldots,c_n)= \int_{[0,1]^n} {\prod_{k=1}^n x_k^{a_k}(1-x_k)^{b_k}\over \prod_{k=2}^n\delta_k({\mathbf x})^{c_k}}{dx_1\ldots dx_n\over\delta_n({\mathbf x})}$ where $$\delta_k(x_1,\ldots,x_n)=1-x_k\delta_{k-1}(x_1,\ldots,x_n)$$. These integrals generalise the integrals found by F. Beukers [Bull. Lond. Math. Soc. 11, 268-272 (1979; Zbl 0421.10023)] and V. N. Sorokin [Mosc. Univ. Math. Bull. 53, No. 1, 53-56 (1983); translation from Vestn. Mosk. Univ., Ser. I 1983, No. 1, 44-47 (1983; Zbl 0508.41014)] in their proofs of the irrationality of $$\zeta(3)$$ and the $$n$$-dimensional integrals of D. V. Vasilev [Dokl. NAN Belarusi 45, No. 5, 36-40 (2001)]. He then shows that these integrals have the invariance properties discovered by G. Rhin and C. Viola in their work on $$\zeta(3)$$ [Acta Arith. 97, 269-293 (2001; Zbl 1004.11042)]. That is, he describes an explicit group of transformations $$G$$ of order 32 such that $$L(g{\mathbf p})/L({\mathbf p})$$ is rational for $$g$$ in $$G$$. For some special choices of the parameters, he finds richer group structures.

##### MSC:
 11M41 Other Dirichlet series and zeta functions 11J72 Irrationality; linear independence over a field 11G55 Polylogarithms and relations with $$K$$-theory
zeta-functions
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##### References:
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