arXiv subjects

Aimin Guo

Publications and source records attributed to Aimin Guo.

5 recordsLinked to original sources

arXiv2026

Normal Behavior and Periodic Points of a Pseudo-Aliquot Map Associated with the Dedekind Psi Function

We study the iteration of (T(n)=\psi(n)-n), where (\psi) 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.

math.NT↗

arXiv2026

Mean Values and Normal Order of a Successive-Iterate Ratio of Dedekind's Arithmetic Function

We obtain asymptotic formulas for the three sums \[ \sum_{n\le x}\frac{\psi(\psi(n))}{\psi(n)},\qquad \sum_{n\le x}\frac{\psi(n)}{\psi(\psi(n))},\qquad \sum_{n\le x}\log\frac{\psi(\psi(n))}{\psi(n)}, \] where $\psi$ denotes Dedekind's arithmetic function. We also determine the normal order of the successive-iterate ratio; more precisely, \[ \frac{\psi(\psi(n))}{\psi(n)} \sim \frac{6e^\gamma}{\pi^2}\log_3 n \] for almost all positive integers $n$.

math.NT↗

arXiv2026

Exceptional sets for compositions involving Euler's function,the divisor-sum function and Dedekind's function

Let $\psi(n)$, $\phi(n)$, and $\sigma(n)$ denote Dedekind's arithmetic function, Euler's totient function, and the sum-of-divisors function, respectively. We study exceptional sets associated with the compositions $\phi(\psi(n))$, $\phi(\sigma(n))$, $\psi(\psi(n))$, and $\psi(\sigma(n))$. For every fixed $c>0$, we obtain quantitative upper bounds for the numbers of integers $n\le x$ satisfying $\phi(\psi(n))\ge cn$ and $\phi(\sigma(n))\ge cn$, thereby refining density results of S\'andor and Dixit and Bhattacharjee, respectively. We further establish quantitative density-zero estimates for the sets of integers $n\le x$ for which $\psi(\psi(n))\le cn$ or $\psi(\sigma(n))\le cn$. 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.

math.NT↗

arXiv2026

On the distribution of $\phi(\psi(n))$ and $\psi(\psi(n))$

Let $\psi(n)$ and $\phi(n)$ denote Dedekind's arithmetic function and Euler's totient function, respectively. We study the distribution of the compositions $\phi(\psi(n))$ and $\psi(\psi(n))$. In particular, we obtain quantitative upper bounds for the exceptional set associated with $\phi(\psi(n))$, thereby refining a density result of S\'andor. We also prove that, for every fixed $c>0$, the set of integers $n\leq x$ satisfying $\psi(\psi(n))\leq cn$ has asymptotic density zero. Our method adapts sieve ideas used by Dixit and Bhattacharjee to compositions involving Dedekind's arithmetic function.

math.NT↗

arXiv2025

On the Composition of the Euler Function and the Dedekind Arithmetic Function

Let $I(n) = \frac{\psi(\phi(n))}{\phi(\psi(n))}$ and $K(n) = \frac{\psi(\phi(n))}{\phi(\phi(n))}$, where $\phi(n)$ is Euler's function and $\psi(n)$ is Dedekind's arithmetic function. We obtain the maximal order of $I(n)$, as well as the average orders of $I(n)$ and $K(n)$. Additionally, we prove a density theorem for both $I(n)$ and $K(n)$.

math.NT↗