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A Hardy space approximation supporting zero-free half-planes for the ζ-function
Abstract
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/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 denotes the half-plane Re(s)>α. We first show that if E belongs to the closure of span(Gk)k≥ 2 in H2(ℂα) for α>1/2, then ζ is zero-free in ℂα. We then use this as the basis for a numerical analysis of the sequence sn = | ∑k=2n μ(k) Gk − E |2α, for 1/2≤ α≤ 1, where |.|α is the norm in H2(ℂα) and μ the Möbius function.
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BibTeX
@misc{manzur2026hardy,
title = {{A Hardy space approximation supporting zero-free half-planes for the $ζ$-function}},
author = {Juan Manzur and Waleed Noor and Gustavo Quintero},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.16097},
eprint = {2606.16097},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.16097},
}
RIS
TY - UNPB
TI - A Hardy space approximation supporting zero-free half-planes for the ζ-function
AU - Juan Manzur
AU - Waleed Noor
AU - Gustavo Quintero
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2606.16097
UR - https://arxiv.org/abs/2606.16097
AB - 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/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 denotes the half-plane Re(s)>α. We first show that if E belongs to the closure of span(G_k)_k≥ 2 in H^2(ℂ_α) for α>1/2, then ζ is zero-free in ℂ_α. We then use this as the basis for a numerical analysis of the sequence s_n = | ∑_k=2^n μ(k) G_k − E |^2_α, for 1/2≤ α≤ 1, where |.|_α is the norm in H^2(ℂ_α) and μ the Möbius function.
ER -