Linear forms in multiple zeta values and multiple integrals
\textit{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 \textit{F. Beukers} [Bull. Lond. Math. Soc. 11, 268-272 (1979; Zbl 0421.10023)] and \textit{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 \textit{D. V. Vasilev} [Dokl. NAN Belarusi 45, No. 5, 36-40 (2001)]. He then shows that these integrals have the invariance properties discovered by \textit{G. Rhin} and \textit{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.
scientific article; zbMATH DE number 1777354