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

TL;DR

This paper explores orthogonality in Hardy spaces related to the Riemann Hypothesis, analyzing the orthogonal complement of a specific function sequence and its implications for the zeros of the zeta function.

Contribution

It introduces new results on the orthogonal complement of a function family in Hardy spaces and links these findings to the distribution of zeta zeros, advancing the Hardy space approach to RH.

Findings

Size and dimension of the orthogonal complement relate to zeta zeros.

The sequence has a complete biorthogonal sequence in Hardy space.

Results suggest implications for the number of zeros if RH fails.

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.