Hilbert's inequality and Witten's zeta-function

In this article we explore a variety of pleasing connections between analysis, number theory, and operator theory, while revisiting a number of beautiful inequalities originating with Hilbert, Hardy and others. We first establish the Hilbert inequality and then apply it to various multiple zeta values. In consequence we obtain the norm of the classical Hilbert matrix, in the process illustrating the interplay of numerical and symbolic computation with classical mathematics.