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

arXiv:2611.03924v1 Announce Type: new Abstract: By a theorem of Bagchi, the Riemann Hypothesis (RH) holds if and only if $E(s) = 1/s$ 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 $\mathbb{C}_\alpha$ is the half-plane $\mathrm{Re}(s) > \alpha$. The formal M\"obius expansion $E = \sum_{k \geq 2} \mu(k) G_k$ diverges on the critical line, by a theorem of B\'aez-Duarte. We study it in the scale of spaces $H^2(\mathbb{C}_\alpha)$, where its convergence forces $\zeta$ to be zero-free in $\mathbb{C}_\alpha$. We prove that the partial sums $S_n$ converge to $E$ in $H^2(\mathbb{C}_\alpha)$ for every $\alpha > 1$, and that a single zero of $\zeta$ in $\mathbb{C}_\alpha$ keeps the best-approximation error $d_n(\alpha)^2$ uniformly away from zero. We express the M\"obius error $s_n(\alpha) = \|S_n - E\|^2_\alpha$ as a series with elementary terms, free of $\zeta$, and compute it for $n \leq 10^6$: $s_n(1/2)$ oscillates without decaying, whereas for $\alpha > 1/2$ it decays at a rate close to $n^{1-2\alpha}$, much faster than the bound known under RH. On the critical line, $d_n(1/2)^2$ behaves like $0.046/\log n$ for $100 \leq n \leq 5120$, in agreement with earlier computations by B\'aez-Duarte, Balazard, Landreau and Saias.

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