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A Hardy space approach to the Nyman-Beurling and B\'aez-Duarte criterion forthe Riemann Hypothesis (RH) was introduced recently in [18] and furtherdeveloped in [13]. It states that the RH holds if and only if a particularsequence of functions $(h_k)_{k\geq 2}$ is complete in the Hardy space $H^2$.This article is concerned with orthogonality questions related to the family$(h_k)_{k\geq 2}$. The first goal is to analyze the orthogonal complement of$\mathcal{N}=\mathrm{span}(h_k)_{k\geq 2}$ in $H^2$. Unbounded Toeplitzoperators on $H^p$ spaces and de Branges-Rovnyak spaces play a central role andour results show that the size and dimension of $\mathcal{N}^\perp$ revealinformation on the zeros of the Riemann $\zeta$-function. The second goal is toshow that $(h_k)_{k\geq 2}$ possesses a complete biorthogonal sequence in$H^2$. We also discuss a folklore conjecture about the number of $\zeta$-zerosif the RH fails.

Orthogonality questions in the Hardy space related to $\zeta$-zeros