A Hardy space approximation supporting zero-free half-planes for the $\zeta$-function
An equivalent version of the B\'aez-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 $G_k(s) = (k^{-s} - k^{-1})\zeta(s)/s,\, k \geq 2$ in the Hardy space \( H^2(\mathbb{C}_{1/2}) \), where $\mathbb{C}_{\alpha}$ denotes the half-plane $\mathrm{Re}(s)>\alpha$. We first show that if $E$ belongs to the closure of span$(G_k)_{k\geq 2}$ in \( H^2(\mathbb{C}_{\alpha}) \) for $\alpha>1/2$, then $\zeta$ is zero-free in $\mathbb{C}_\alpha$. We then use this as the basis for a numerical analysis of the sequence \[ s_n = \left\| \sum_{k=2}^{n} \mu(k) G_k - E \right\|^2_\alpha, \] for $1/2\leq \alpha \leq 1$, where $\left\|.\right\|_\alpha$ is the norm in $H^2(\mathbb{C}_{\alpha})$ and $\mu$ the M\"obius function.