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

Abstract

A Hardy space approach to the Nyman-Beurling and B\'aez-Duarte criterion for the Riemann Hypothesis (RH) was introduced recently in [18] and further developed in [13]. It states that the RH holds if and only if a particular sequence 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 Toeplitz operators on $H^p$ spaces and de Branges-Rovnyak spaces play a central role and our results show that the size and dimension of $\mathcal{N}^\perp$ reveal information on the zeros of the Riemann $\zeta$-function. The second goal is to show that $(h_k)_{k\geq 2}$ possesses a complete biorthogonal sequence in $H^2$. We also discuss a folklore conjecture about the number of $\zeta$-zeros if the RH fails.