AI 中文总结
该数论研究针对欧拉函数、除数和函数与戴德金函数的四类复合函数，推导了相关例外集的定量上界与密度为零的估计，改进了已有研究成果。
AI 中文摘要
令ψ(n)、φ(n)和σ(n)分别表示戴德金算术函数、欧拉 totient 函数与除数和函数。我们研究复合函数φ(ψ(n))、φ(σ(n))、ψ(ψ(n))和ψ(σ(n))相关的例外集。对任意固定的c>0，我们得到满足φ(ψ(n))≥cn和φ(σ(n))≥cn的整数n≤x的数量的定量上界，分别改进了Sándor及Dixit与Bhattacharjee的密度结果。我们还建立了满足ψ(ψ(n))≤cn或ψ(σ(n))≤cn的整数n≤x的集合的定量密度为零的估计。更一般地，我们允许前两个问题中的固定阈值被替换为满足温和增长条件的非递减函数对应的阈值。
英文摘要
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 \(n\le x\) satisfying \(ϕ(ψ(n))\ge cn\) and \(ϕ(σ(n))\ge cn\), thereby refining density results of Sándor and Dixit and Bhattacharjee, respectively. We further establish quantitative density-zero estimates for the sets of integers \(n\le x\) for which \(ψ(ψ(n))\le cn\) or \(ψ(σ(n))\le cn\). 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.
