Orthogonality questions in the Hardy space related to $ζ$-zeros
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866910686109499392 |
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| author | Calderaro, Francisco Manzur, Juan Noor, Waleed Santos, Charles |
| author_facet | Calderaro, Francisco Manzur, Juan Noor, Waleed Santos, Charles |
| contents | A Hardy space approach to the Nyman-Beurling and Báez-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 $ζ$-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 $ζ$-zeros if the RH fails. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_05030 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Orthogonality questions in the Hardy space related to $ζ$-zeros Calderaro, Francisco Manzur, Juan Noor, Waleed Santos, Charles Functional Analysis A Hardy space approach to the Nyman-Beurling and Báez-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 $ζ$-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 $ζ$-zeros if the RH fails. |
| title | Orthogonality questions in the Hardy space related to $ζ$-zeros |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2203.05030 |