Multiplicative Colombeau algebras and the Nyman--Beurling criterion for the Riemann hypothesis
Abstract
This paper establishes an equivalence between the Riemann hypothesis and the association, together with uniform L2-boundedness, of a moderate net in a Colombeau-type algebra built from polynomially damped Báez-Duarte sums. The regularization is performed by multiplicative (Mellin) convolution, which respects the dilation symmetry of the Beurling functions and guarantees that every approximant lies in the L2-closure of the Beurling space. The equivalence is unconditional under the Riemann hypothesis: it uses only the qualitative convergence of Báez-Duarte, Mazur's theorem, and the classical Nyman--Beurling criterion. As a separate quantitative refinement, we prove that under two additional hypotheses on the non-trivial zeros of ζ (simplicity and separation), the damping error admits a power-law bound with an explicit constant. This refinement is independent of the equivalence and is not used in its proof. The exponential damping e−keps2 is discussed as an open problem.
Related
- Spectra of Beurling algebras of abelian groups
- Non-vanishing mod p of derived Hecke algebras of multiplicative groups over number fields, II
- Commutator identities, Lie product identities, and Multiplicative Lie algebras
- Explicit formula for the discrete Laplace transform of the Möbius function, related special functions, and a criterion for the Riemann hypothesis
- Hochschild theory of multiplicative sequences of algebras and coalgebra measurings
BibTeX
@misc{cruz2026multiplicative,
title = {{Multiplicative Colombeau algebras and the Nyman--Beurling criterion for the Riemann hypothesis}},
author = {Amaury Alvarez Cruz and Esteban A Alvarez Gutierrez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.22562},
eprint = {2606.22562},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.22562},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multiplicative Colombeau algebras and the Nyman--Beurling criterion for the Riemann hypothesis
AU - Amaury Alvarez Cruz
AU - Esteban A Alvarez Gutierrez
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2606.22562
UR - https://arxiv.org/abs/2606.22562
AB - This paper establishes an equivalence between the Riemann hypothesis and the association, together with uniform L^2-boundedness, of a moderate net in a Colombeau-type algebra built from polynomially damped Báez-Duarte sums. The regularization is performed by multiplicative (Mellin) convolution, which respects the dilation symmetry of the Beurling functions and guarantees that every approximant lies in the L^2-closure of the Beurling space. The equivalence is unconditional under the Riemann hypothesis: it uses only the qualitative convergence of Báez-Duarte, Mazur's theorem, and the classical Nyman--Beurling criterion. As a separate quantitative refinement, we prove that under two additional hypotheses on the non-trivial zeros of ζ (simplicity and separation), the damping error admits a power-law bound with an explicit constant. This refinement is independent of the equivalence and is not used in its proof. The exponential damping e^−keps^2 is discussed as an open problem.
ER -