Well-poised hypergeometric service for diophantine problems of zeta values

It is explained how the classical concept of well-poised hypergeometric series and integrals becomes crucial in studying arithmetic properties of the values of Riemann’s zeta function. By these well-poised means we obtain: (1) a permutation group for linear forms in 1 and ζ(4)=π 4 /90 yielding a conditional upper bound for the irrationality measure of ζ(4); (2) a second-order Apéry-like recursion for ζ(4) and some low-order recursions for linear forms in odd zeta values; (3) a rich permutation group for a family of certain Euler-type multiple integrals that generalize so-called Beukers’ integrals for ζ(2) and ζ(3).