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

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 $G_k(s) = (k^{-s} - k^{-1})ζ(s)/s$, $k \geq 2$, in the Hardy space $H^2(\mathbb{C}_{1/2})$, where $\mathbb{C}_α$ is the half-plane $\mathrm{Re}(s) > α$. The formal Möbius expansion $E = \sum_{k \geq 2} μ(k) G_k$ diverges on the critical line, by a theorem of Báez-Duarte. We study it in the scale of spaces $H^2(\mathbb{C}_α)$, where its convergence forces $ζ$ to be zero-free in $\mathbb{C}_α$. We prove that the partial sums $S_n$ converge to $E$ in $H^2(\mathbb{C}_α)$ for every $α> 1$, and that a single zero of $ζ$ in $\mathbb{C}_α$ keeps the best-approximation error $d_n(α)^2$ uniformly away from zero. We express the Möbius error $s_n(α) = \|S_n - E\|^2_α$ as a series with elementary terms, free of $ζ$, and compute it for $n \leq 10^6$: $s_n(1/2)$ oscillates without decaying, whereas for $α> 1/2$ it decays at a rate close to $n^{1-2α}$, 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 Báez-Duarte, Balazard, Landreau and Saias.