论文标题
与$ζ$ -Zeros有关的顽固空间中的正交性问题
Orthogonality questions in the Hardy space related to $ζ$-zeros
论文作者
论文摘要
最近在[18]中引入了针对Riemann假设（RH）的Nyman-Beberling和Báez-Duarte标准的强大空间方法，并在[13]中进一步开发。它指出，仅当特定的函数序列$（h_k）_ {k \ geq 2} $才完成时，rh才能保持在hardy space $ h^2 $中。本文涉及与家庭$（h_k）_ {k \ geq 2} $有关的正交问题。第一个目标是分析$ \ mathcal {n} = \ mathrm {span}（h_k）_ {k \ geq 2} $ in $ h^2 $中的正交补充。 $ h^p $空间上的无界Toeplitz运营商和de Branges-Rovnyak空间起着核心作用，我们的结果表明，$ \ Mathcal {n}^\ perp $的大小和维度揭示了Riemann $ζ$ function的零的信息。第二个目标是证明$（h_k）_ {k \ geq 2} $在$ h^2 $中具有完整的生物表达序列。我们还讨论了一个民间传说的猜想，内容涉及$ζ$ -Zeros如果失败。
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.
