On certain Gram matrices and their associated series

We derive formulae for Gram matrices arising in the Nyman--Beurling reformulation of the Riemann hypothesis. The development naturally leads upon series of the form $S(x) = \sum_{n\ge 1} R(nx)$ and their reciprocity relations. We give integral representations of these series; and we present decompositions of the quadratic forms associated with the Gram matrices along with a discussion of the components' properties.