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 $\zeta \left(4\right)={\pi }^4/90$ yielding a conditional upper bound for the irrationality measure of $\zeta \left(4\right)$; (2) a second-order Apéry-like recursion for $\zeta \left(4\right)$ and some low-order recursions for linear forms in odd zeta values; (3) a rich permutation group for a family...