A Hardy space approximation supporting zero-free half-planes for the $\zeta$-function

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 $E(s)=1\mathrm{/}s$ belongs to the closed linear span of $G_k(s)=(k^{-s}-k^{-1})\zeta (s)\mathrm{/}s,k\ge 2$ in the Hardy space $H^2({\mathbb{C}}_{1\mathrm{/}2})$, where ${\mathbb{C}}_{\alpha }$ denotes the half-plane $\mathrm{R}\mathrm{e}(s)>\alpha$. We first show that if $E$ belongs to the closure of span$(G_k{)}_{k\ge 2}$ in $H^2({\mathbb{C}}_{\alpha })$ for $\alpha >1\mathrm{/}2$, then $\zeta$ is zero-free in ${\mathbb{C}}_{\alpha }$. We then use this as the basis for a numerical analysis of the sequence $$s_n={\parallel \sum _{k=2}^n\mu (k)G_k-E\parallel }_{\alpha }^2,$$ for $1\mathrm{/}2\le \alpha \le 1$, where ${\parallel \mathrm{.}\parallel }_{\alpha }$ is the norm in $H^2({\mathbb{C}}_{\alpha })$ and $\mu$ the Möbius function.