  A Hardy space approach to the Nyman-Beurling and B\'aez-Duarte criterion for
the Riemann Hypothesis (RH) was introduced by Noor [9]. It states that the RH
holds if and only if the linear manifold $\mathcal{N}$ generated by a sequence
of functions $(h_k)_{k\geq 2}$ is dense in the Hardy space $H^2$. In this
article we consider the following three questions: $\mathbf{A.}$ How small is
the orthogonal complement of $\mathcal{N}$? $\mathbf{B.}$ Is the closure of
$\mathcal{N}$ a shift-invariant subspace of $H^2$? $\mathbf{C.}$ Is
$\mathcal{N}$ dense in $H^2$ under a weaker topology?. We use tools from local
Dirichlet spaces, the de Branges-Rovnyak spaces and the Smirnov class to
address these questions.
