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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:
 [1] Apéry, R., Irrationalité de ζ(2) et ζ(3), Astérisque, 61, 11-13, (1979) · Zbl 0401.10049 [2] Beukers, F., A note on the irrationality of ζ(2) and ζ(3), Bull. London math. soc., 11, 3, 268-272, (1979) · Zbl 0421.10023 [3] S. Fischler, Groupes de Rhin-Viola et intégrales multiples, J. Théor. Nombres Bordeaux, soumis · Zbl 1074.11040 [4] Kontsevich, M.; Zagier, D., Periods, (), 771-808 · Zbl 1039.11002 [5] Rhin, G.; Viola, C., On a permutation group related to ζ(2), Acta arith., 77, 1, 23-56, (1996) · Zbl 0864.11037 [6] Rhin, G.; Viola, C., The group structure for ζ(3), Acta arith., 97, 3, 269-293, (2001) · Zbl 1004.11042 [7] Sorokin, V.N., A transcendence measure for π2, Sb. math., 187, 12, 1819-1852, (1996) · Zbl 0876.11035 [8] Sorokin, V.N., Apéry’s theorem, Moscow univ. math. bull., 53, 3, 48-52, (1998) · Zbl 1061.11501 [9] Vasilyev, D.V., Some formulas for Riemann zeta-function at integer points, Moscow univ. math. bull., 51, 1, 41-43, (1996) [10] Vasilyev, D.V., On small linear forms for the values of the Riemann zeta-function at odd integers, Doklady NAN belarusi (reports of the belarus national Academy of sciences), 45, 5, 36-40, (2001), (en russe) [11] Waldschmidt, M., Valeurs zêta multiples : une introduction, J. théor. nombres Bordeaux, 12, 2, 581-595, (2000) · Zbl 0976.11037 [12] Zlobin, S., Integrals represented as linear forms in generalized polylogarithms, Mat. zametki, 71, 5, 782-787, (2002), (en russe)
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