  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.
