PreprintCC BY 4.0arXiv
A Hardy-space operator acting on ζ-zeros
Abstract
A Hardy space approach to the Nyman-Beurling and Báez-Duarte criterion for the Riemann Hypothesis (RH) was introduced in [7]. It states that the RH holds if and only if the span of a particular sequence of functions (hk)k≥2, denoted 𝒩, is dense in the Hardy space H2. In this note we construct a bounded operator T on H2 such that if the closure of 𝒩 is T-invariant, then either the RH holds or there exists an infinite progression of ζ-zeros. In particular, this yields an operator-theoretic criterion for the Riemann hypothesis as an invariant subspace question.
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BibTeX
@misc{manzur2026hardyspace,
title = {{A Hardy-space operator acting on $ζ$-zeros}},
author = {Juan Manzur and Waleed Noor},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03568},
eprint = {2610.03568},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03568},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Hardy-space operator acting on ζ-zeros
AU - Juan Manzur
AU - Waleed Noor
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03568
UR - https://arxiv.org/abs/2610.03568
AB - A Hardy space approach to the Nyman-Beurling and Báez-Duarte criterion for the Riemann Hypothesis (RH) was introduced in [7]. It states that the RH holds if and only if the span of a particular sequence of functions (h_k)_k≥2, denoted 𝒩, is dense in the Hardy space H^2. In this note we construct a bounded operator T on H^2 such that if the closure of 𝒩 is T-invariant, then either the RH holds or there exists an infinite progression of ζ-zeros. In particular, this yields an operator-theoretic criterion for the Riemann hypothesis as an invariant subspace question.
ER -