PreprintCC BY 4.0arXiv
Normal Behavior and Periodic Points of a Pseudo-Aliquot Map Associated with the Dedekind Psi Function
Abstract
We study the iteration of (T(n)=ψ(n)-n), where (ψ) is the Dedekind psi function. We show that, for almost all integers, successive iterates satisfy strong asymptotic growth relations governed by (T(n)/n). As an application, we prove that for every fixed (\ell\ge1), the set of integers satisfying (T^{(\ell)}(n)=n) has asymptotic density zero, and hence so does the set of periodic points of any fixed period.
Related
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BibTeX
@misc{guo2026normal,
title = {{Normal Behavior and Periodic Points of a Pseudo-Aliquot Map Associated with the Dedekind Psi Function}},
author = {Aimin Guo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06332},
eprint = {2609.06332},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06332},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Normal Behavior and Periodic Points of a Pseudo-Aliquot Map Associated with the Dedekind Psi Function
AU - Aimin Guo
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06332
UR - https://arxiv.org/abs/2609.06332
AB - We study the iteration of (T(n)=ψ(n)-n), where (ψ) is the Dedekind psi function. We show that, for almost all integers, successive iterates satisfy strong asymptotic growth relations governed by (T(n)/n). As an application, we prove that for every fixed (\ell\ge1), the set of integers satisfying (T^{(\ell)}(n)=n) has asymptotic density zero, and hence so does the set of periodic points of any fixed period.
ER -