Eden–Staudacher and Beisert–Eden–Staudacher equations in the $\mathcal N=4$ supersymmetric gauge theory

We investigate the Eden–Staudacher and Beisert–Eden–Staudacher equations for the anomalous dimension of twist-$2$ operators at a large spin $s$ in the $\mathcal{N}{=}4$ supersymmetric gauge theory. We reduce these equations to a set of linear algebraic equations and calculate their kernels analytically. We demonstrate that in the perturbation theory, the anomalous dimension is a sum of products of the Euler functions $\zeta(k)$ having the maximum transcendentality property. We also show that at a large coupling, the "singular" solution of the Beisert–Eden–Staudacher equation reproduces the anomalous dimension constants predicted from the string side of the AdS/CFT correspondence.