On certain Gram matrices and their associated series

TL;DR

This paper explores Gram matrices linked to the Nyman--Beurling reformulation of the Riemann hypothesis, providing new formulae, integral representations, and decompositions of related series and quadratic forms.

Contribution

It introduces novel formulae and decompositions for Gram matrices and series relevant to the Nyman--Beurling approach to the Riemann hypothesis.

Findings

Derived explicit formulae for Gram matrices.

Provided integral representations of series of the form S(x).

Analyzed decompositions of quadratic forms associated with Gram matrices.

Abstract

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.