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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
Keywords:
zeta-functions
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[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
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[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
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[8] Sorokin, V.N., Apéry’s theorem, Moscow univ. math. bull., 53, 3, 48-52, (1998) · Zbl 1061.11501
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[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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