PreprintCC BY 4.0arXiv
Exceptional sets for compositions involving Euler's function,the divisor-sum function and Dedekind's function
Abstract
Let ψ(n), ϕ(n), and σ(n) denote Dedekind's arithmetic function, Euler's totient function, and the sum-of-divisors function, respectively. We study exceptional sets associated with the compositions ϕ(ψ(n)), ϕ(σ(n)), ψ(ψ(n)), and ψ(σ(n)). For every fixed c>0, we obtain quantitative upper bounds for the numbers of integers n≤ x satisfying ϕ(ψ(n))≥ cn and ϕ(σ(n))≥ cn, thereby refining density results of Sándor and Dixit and Bhattacharjee, respectively. We further establish quantitative density-zero estimates for the sets of integers n≤ x for which ψ(ψ(n))≤ cn or ψ(σ(n))≤ cn. More generally, we allow the fixed thresholds in the first two problems to be replaced by thresholds involving non-decreasing functions subject to mild growth conditions.
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BibTeX
@misc{guo2026exceptional,
title = {{Exceptional sets for compositions involving Euler's function,the divisor-sum function and Dedekind's function}},
author = {Aimin Guo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.19972},
eprint = {2608.19972},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.19972},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exceptional sets for compositions involving Euler's function,the divisor-sum function and Dedekind's function
AU - Aimin Guo
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2608.19972
UR - https://arxiv.org/abs/2608.19972
AB - Let ψ(n), ϕ(n), and σ(n) denote Dedekind's arithmetic function, Euler's totient function, and the sum-of-divisors function, respectively. We study exceptional sets associated with the compositions ϕ(ψ(n)), ϕ(σ(n)), ψ(ψ(n)), and ψ(σ(n)). For every fixed c>0, we obtain quantitative upper bounds for the numbers of integers n≤ x satisfying ϕ(ψ(n))≥ cn and ϕ(σ(n))≥ cn, thereby refining density results of Sándor and Dixit and Bhattacharjee, respectively. We further establish quantitative density-zero estimates for the sets of integers n≤ x for which ψ(ψ(n))≤ cn or ψ(σ(n))≤ cn. More generally, we allow the fixed thresholds in the first two problems to be replaced by thresholds involving non-decreasing functions subject to mild growth conditions.
ER -