[Paper Review] On certain arithmetic functions involving exponential divisors

László Tóth|arXiv (Cornell University)|Oct 9, 2006

Analytic Number Theory Research13 references18 citations

TL;DR

This paper investigates arithmetic functions involving exponential divisors and exponentially coprime integers, introducing and analyzing functions such as $\phi^{(e)}(n)$, $\tilde{\sigma}(n)$, and $\tilde{P}(n)$, with key results on their asymptotic averages and maximal orders. It establishes precise asymptotic formulas for their summatory functions and derives the maximal order of $\tilde{P}(n)$, showing $\limsup_{n\to\infty} \frac{\tilde{P}(n)}{n\log\log n} = \frac{6}{\pi^2}e^{\gamma}$. The work generalizes prior results on $k$-free and squarefree integers using Dirichlet series and multiplicative function techniques.

ABSTRACT

The integer $d$ is called an exponential divisor of $n=\prod_{i=1}^r p_i^{a_i}>1$ if $d=\prod_{i=1}^r p_i^{c_i}$, where $c_i \mid a_i$ for every $1\le i \le r$. The integers $n=\prod_{i=1}^r p_i^{a_i}, m=\prod_{i=1}^r p_i^{b_i}>1$ having the same prime factors are called exponentially coprime if $(a_i,b_i)=1$ for every $1\le i\le r$. In the paper we investigate asymptotic properties of certain arithmetic functions involving exponential divisors and exponentially coprime integers.

Motivation & Objective

• To investigate the asymptotic behavior of arithmetic functions involving exponential divisors, including $\phi^{(e)}(n)$, $\tilde{\sigma}(n)$, and $\tilde{P}(n)$.
• To generalize results on $k$-free and exponentially squarefree integers by studying the summatory function $\sum_{n\leq x} q_k^{(e)}(n)$.
• To determine the maximal order of $\tilde{P}(n)$ and establish precise asymptotic formulas for the average orders of $\phi^{(e)}(n)$, $\tilde{\sigma}(n)$, and $\tilde{P}(n)$.
• To explore open problems related to the order of $\varphi_e(n)$, the minimal order of $\tilde{\sigma}(n)$, and the maximal order of $\omega(\phi^{(e)}(n))$.

Proposed method

• The paper uses multiplicative function theory and Dirichlet series to analyze $\tilde{P}(n)$, deriving its Dirichlet series representation $\sum_{n=1}^\infty \frac{\tilde{P}(n)}{n^s} = \frac{\zeta(s-1)\zeta(2s-1)}{\zeta(3s-2)} W(s)$ with $W(s)$ absolutely convergent for $\operatorname{Re}(s) > 3/4$.
• It applies the general result of Sita Ramaiah and Suryanarayana to estimate $\sum_{n\leq x} q_k^{(e)}(n)$, relying on Walfisz's estimate for $k$-free integers.
• For $\sum_{n\leq x} (\tilde{\sigma}(n))^u$, the method involves partial summation and bounding $g(n) = (\tilde{\sigma}(n)/n)^u$ with $0 < g(n) \leq 1$, applying Lemma 2 with $k=2$, $\beta = u$.
• The maximal order of $\tilde{P}(n)$ is derived using a general result of Tóth and Wirsing, applying it to $f(n) = \tilde{P}(n)/n$ with $f(p^2) = 1 + 1/p$ and $\rho(p) = 1 + 1/p$.
• The function $\phi^{(e)}(n)$ is shown to be multiplicative and dependent only on the cubfull kernel of $n$, with $\phi^{(e)}(p^a) = \phi(a)$, enabling the use of known techniques for multiplicative functions.
• The paper uses partial summation and estimates on $\sum_{mn^2 \leq x} mn$ to derive the asymptotic for $\sum_{n\leq x} \tilde{P}(n)$, drawing on results from Pétermann and Wu.

Experimental results

Research questions

• RQ1 What is the average order of $\phi^{(e)}(n)$, and what are the constants in its asymptotic expansion?
• RQ2 What is the asymptotic behavior of $\sum_{n\leq x} (\tilde{\sigma}(n))^u$ for $u > 1/3$, and what is the error term?
• RQ3 What is the maximal order of $\tilde{P}(n)$, and how does it relate to $n\log\log n$?
• RQ4 What is the asymptotic formula for $\sum_{n\leq x} \tilde{P}(n)$, and what is the constant $C_4$ in its leading term?
• RQ5 What is the maximal order of $\Omega(\phi^{(e)}(n))$, and for which $n$ is it attained?

Key findings

• The sum $\sum_{n\leq x} \phi^{(e)}(n)$ has the asymptotic expansion $C_1 x + C_2 x^{1/3} + O(x^{1/5 + \varepsilon})$ for every $\varepsilon > 0$, with explicit constants $C_1$ and $C_2$ defined via Euler products.
• For $u > 1/3$, the sum $\sum_{n\leq x} (\tilde{\sigma}(n))^u$ satisfies $C_3 x^{u+1} + O(x^{u+1/2} \delta(x))$, where $\delta(x) = \exp(-A(\log x)^{3/5} (\log\log x)^{-1/5})$ and $C_3$ is given by an Euler product.
• The sum $\sum_{n\leq x} \tilde{P}(n)$ is asymptotically $C_4 x^2 + O(x (\log x)^{5/3})$, with $C_4 = \frac{1}{2} \prod_p \left(1 + \sum_{a=2}^\infty \frac{\tilde{P}(p^a) - p \tilde{P}(p^{a-1})}{p^{2a}} \right)$.
• The maximal order of $\tilde{P}(n)$ satisfies $\limsup_{n\to\infty} \frac{\tilde{P}(n)}{n \log\log n} = \frac{6}{\pi^2} e^{\gamma}$, where $\gamma$ is Euler's constant.
• The maximal order of $\Omega(\phi^{(e)}(n))$ is $\frac{2}{5} \frac{\log n}{\log\log n}$, achieved when $n_k = (p_1 \cdots p_k)^5$ for the first $k$ primes.
• The paper generalizes Wu's result on exponentially squarefree integers by deriving an asymptotic for $\sum_{n\leq x} q_k^{(e)}(n)$, valid for $k \geq 2$, using techniques from Sita Ramaiah and Suryanarayana.