A Hardy space approximation supporting zero-free half-planes for the $\hat{I}\P$-function

Abstract

An equivalent version of the Báez-Duarte criterion \cite{baez} for the Riemann Hypothesis (RH) by Bagchi states that the RH holds true if and only if the function $E(s)=1/s$ belongs to the closed linear span of $G_k(s)=(k^{-s}-k^{-1})\hat{I}\P (s)/s,k\ge 2$ in the Hardy space \( H^2(\mathbb{C}_{1/2}) \), where ${\mathbb{C}}_{\hat{I}}\pm$ denotes the half-plane $\mathrm{Re}(s)>\hat{I}\pm$. We first show that if $E$ belongs to the closure of span$(G_k{)}_{k\ge 2}$ in \( H^2(\mathbb{C}_α) \) for $\hat{I}\pm >1/2$, then $\hat{I}\P$ is zero-free in ${\mathbb{C}}_{\hat{I}}\pm$. We then use this as the basis for a numerical analysis of the sequence \[ s_n = \left\| \sum_{k=2}^{n} μ(k) G_k - E \right\|^2_α, \] for $1/2\le \hat{I}\pm \le 1$, where ${\Vert .\Vert }_{\hat{I}}\pm$ is the norm in $H^2({\mathbb{C}}_{\hat{I}}\pm )$ and $\hat{I}¼$ the Möbius function.