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 Gk(s) = (k−s − k−1)ζ(s)/s, k ≥ 2, in the Hardy space H2(ℂ1/2), where ℂα is the half-plane Re(s) > α. The formal Möbius expansion E = ∑k ≥ 2 μ(k) Gk diverges on the critical line, by a theorem of Báez-Duarte. We study it in the scale of spaces H2(ℂα), where its convergence forces ζ to be zero-free in ℂα. We prove that the partial sums Sn converge to E in H2(ℂα) for every α> 1, and that a single zero of ζ in ℂα keeps the best-approximation error dn(α)2 uniformly away from zero. We express the Möbius error sn(α) = |Sn − E|2α as a series with elementary terms, free of ζ, and compute it for n ≤ 106: sn(1/2) oscillates without decaying, whereas for α> 1/2 it decays at a rate close to n1−2α, much faster than the bound known under RH. On the critical line, dn(1/2)2 behaves like 0.046/log n for 100 ≤ n ≤ 5120, in agreement with earlier computations by Báez-Duarte, Balazard, Landreau and Saias.
Related
- The Skew Commutators of Toeplitz and Hankel operators in vector-valued Hardy Space
- Numerical range and invariant subspaces of weighted composition operators on the Hardy space of Dirichlet series
- Local Nevanlinna embedding for Hardy space of Dirichlet series below critical exponent
- Projections and minimal invariant subspaces in the Hardy space over the bidisk
- A Hardy-space operator acting on ζ-zeros
BibTeX
@misc{manzur2026hardy,
title = {{A Hardy space approximation supporting zero-free half-planes for the $ζ$-function}},
author = {Juan Manzur and Gustavo Quintero},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03924},
eprint = {2611.03924},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03924},
}
RIS
TY - UNPB
TI - A Hardy space approximation supporting zero-free half-planes for the ζ-function
AU - Juan Manzur
AU - Gustavo Quintero
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03924
UR - https://arxiv.org/abs/2611.03924
AB - 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 ≥ 2, in the Hardy space H^2(ℂ_1/2), where ℂ_α is the half-plane Re(s) > α. The formal Möbius expansion E = ∑_k ≥ 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(ℂ_α), where its convergence forces ζ to be zero-free in ℂ_α. We prove that the partial sums S_n converge to E in H^2(ℂ_α) for every α> 1, and that a single zero of ζ in 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 ≤ 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 ≤ n ≤ 5120, in agreement with earlier computations by Báez-Duarte, Balazard, Landreau and Saias.
ER -