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.