---
abstract: 'The integers $n=\prod_{i=1}^r p_i^{a_i}$ and $m=\prod_{i=1}^r p_i^{b_i}$ having the same prime factors are called exponentially coprime if $(a_i,b_i)=1$ for every $1\le i\le r$. We estimate the number of pairs of exponentially coprime integers $n,m\le x$ having the prime factors $p_1,...,p_r$ and show that the asymptotic density of pairs of exponentially coprime integers having $r$ fixed prime divisors is $(\zeta(2))^{-r}$.'
author:
- '[László Tóth]{} (Pécs, Hungary)'
date: 'Pure Math. Appl. (PU.M.A.), 15 (2004), 343-348'
title: '**On exponentially coprime integers**'
---

=6.truein =9.truein =-.5truein =-.8truein

\[section\] \[section\] \[section\]

Mathematics Subject Classification (2000): 11A05, 11A25, 11N37

[**1. Introduction**]{}

Let $n>1$ be an integer of canonical form $n=\prod_{i=1}^r p_i^{a_i}$. The integer $d$ is called an [*exponential divisor*]{} of $n$ if $d=\prod_{i=1}^r p_i^{c_i}$, where $c_i | a_i$ for every $1\le i \le r$, notation: $d|_e n$. By convention $1|_e 1$. This notion was introduced by [M. V. Subbarao]{} [@Su72]. The smallest exponential divisor of $n>1$ is its squarefree kernel $\kappa(n):=\prod_{i=1}^r p_i$.

Let $\tau^{(e)}(n)= \sum_{d|_e n} 1$ and $\sigma^{(e)}(n)=\sum_{d|_e n} d$ denote the number and the sum of exponential divisors of $n$, respectively. Properties of these functions were investigated by several authors, see [@FaSu89], [@KaSu2003], [@PeWu97], [@SmWu97], [@Su72], [@Wu95].

Two integers $n,m >1$ have common exponential divisors iff they have the same prime factors and for $n=\prod_{i=1}^r p_i^{a_i}$, $m=\prod_{i=1}^r p_i^{b_i}$, $a_i,b_i\ge 1$ ($1\le i\le r$), the [*greatest common exponential divisor*]{} of $n$ and $m$ is $$(n,m)_e:=\prod_{i=1}^r p_i^{(a_i,b_i)}.$$

Here $(1,1)_e=1$ by convention and $(1,m)_e$ does not exist for $m>1$.

The integers $n,m >1$ are called [*exponentially coprime*]{}, if they have the same prime factors and $(a_i,b_i)=1$ for every $1\le i\le r$, with the notation of above. In this case $(n,m)_e=\prod_{i=1}^r p_i$. $1$ and $1$ are considered to be exponentially coprime. $1$ and $m>1$ are not exponentially coprime. Exponentially coprime integers were introduced by [J. Sándor]{} [@Sa96].

Let $p_i$ ($1\le i\le r$) be fixed distinct primes and let $P^{(e)}(p_1,...,p_r;x)$ denote the number of pairs $\langle n,m \rangle$ of exponentially coprime integers such that $\kappa(n)=\kappa(m)=\prod_{i=1}^r p_i$ and $n,m\le x$.

In this note we estimate $P^{(e)}(p_1,...,p_r;x)$ and show that the asymptotic density of pairs of exponentially coprime integers having $r$ fixed prime divisors is $(\zeta(2))^{-r}$.

As an open problem we formulate the following: What can be said on the asymptotic density of pairs of exponentially coprime integers if their prime divisors are not fixed ?

For a real $x\ge 1$ and an integer $n\ge 1$ consider the Legendre-type function $L^{(e)}(x,n)$ defined as the number of integers $k\le x$ such that $k$ and $n$ are exponentially coprime.

The following estimate holds:

Let $N(p_1,...,p_r;x)$ denote the number of integers $n\le x$ having the kernel $\kappa(n)=p_1\cdots p_r$. Taking $a_1=\cdots =a_r=1$ we obtain from Theorem 1 the following known estimate, cf. for ex. [@Te95], Ch. III.5 regarding integers free of large prime factors.

The proofs of Theorems 1 and 2 are by induction on $r$, while Corollary 3 follows from Theorem 2 and Corollary 1. First we prove the following lemma.

We will use the well-known estimate: if $s\ge 0$, then $$\phi_s(z,a):=\sum_{n\le z \atop{(n,a)=1}} n^s =\frac{z^{s+1}\phi(a)}{(s+1)a}+O(z^s\theta(a)), \leqno(5)$$ uniformly for $z\ge 1$ and $a\ge 1$.

Induction on $r$. For $r=1$ (4) follows from (5) applied for $s=0$. Suppose formula (4) is valid for $r-1$ and prove it for $r$. $$\sum_{k_1t_1+\cdots +k_rt_r\le z \atop{(k_1,a_1)=\cdots =(k_r,a_r)=1 \atop{k_1,...,k_r\ge 1}}} 1 =
\sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{(k_r,a_r)=1 \atop{k_r\ge 1}}}
\sum_{k_1t_1+\cdots +k_{r-1}t_{r-1}\le z-k_rt_r \atop{(k_1,a_1)=\cdots =(k_{r-1},a_{r-1})=1 \atop{k_1,...,k_{r-1}\ge 1}}} 1$$ $$=\sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{(k_r,a_r)=1 \atop{k_r\ge 1}}}
\left(\frac1{(r-1)!} \left( \prod_{i=1}^{r-1} \frac{\phi(a_i)}{a_i t_i}\right)
(z-k_rt_r)^{r-1} + O\left( z^{r-2} \sum_{i=1}^{r-1} \theta(a_i) \right) \right)$$ $$= \frac1{(r-1)!} \left( \prod_{i=1}^{r-1} \frac{\phi(a_i)}{a_i t_i}\right)
\sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{(k_r,a_r)=1 \atop{k_r\ge 1}}} (z-k_rt_r)^{r-1}+
O\left( z^{r-1} \sum_{i=1}^{r-1} \theta(a_i) \right).$$ Using the binomial formula and estimate (5) the sum appearing here is $$\sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} z^{r-1-j} t_r^j \sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{(k_r,a_r)=1 \atop{k_r\ge 1}}} k_r^j$$ $$= \sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} z^{r-1-j} t_r^j \left( \frac{(z-t_1-\cdots -t_{r-1})^{j+1}\phi(a_r)}{(j+1)t_r^{j+1} a_r} +O(z^j \theta(a_r))\right)$$ $$=\frac{\phi(a_r)}{t_ra_r} z^r \sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} \frac1{j+1} + O(  z^{r-1}\theta(a_r))$$ $$=\frac{\phi(a_r)}{rt_ra_r} z^r + O(z^{r-1}\theta(a_r))$$ and the proof is complete.

Apply Lemma 1 for $z=\log x$, $t_1=\log p_1,...,t_r=\log p_r$.

In order to prove Theorem 2 we need

Induction on $r$, similar to the proof of Lemma 1. We use the well-known estimate: let $s\ge -1$ be a real number, then for $z\ge 3$, $$\sum_{n\le z} \phi(n)n^s =\frac{z^{s+2}}{(s+2)\zeta(2)} + O(z^{s+1}\log z). \leqno(7)$$

For $r=1$ (6) follows from (7) applied for $s=-1$. Suppose formula (6) is valid for $r-1$ and prove it for $r$. $$\sum_{k_1t_1+\cdots +k_rt_r\le z \atop{k_1,...,k_r\ge 1}} \prod_{i=1}^r \frac{\phi(k_i)}{k_i} = \sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{k_r\ge 1}} \frac{\phi(k_r)}{k_r}
\sum_{k_1t_1+\cdots +k_{r-1}t_{r-1}\le z-k_rt_r \atop{k_1,...,k_{r-1}\ge 1}} \prod_{i=1}^{r-1} \frac{\phi(k_i)}{k_i}$$ $$=\sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{k_r\ge 1}} \frac{\phi(k_r)}{k_r}
\left(\frac1{(r-1)! (\zeta(2))^{r-1}} \left( \prod_{i=1}^{r-1} \frac1{t_i} \right)
(z-k_rt_r)^{r-1} + O\left( z^{r-2} \log z\right) \right)$$ $$= \frac1{(r-1)! (\zeta(2))^{r-1}} \left( \prod_{i=1}^{r-1} \frac1{t_i}\right)
\sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{k_r\ge 1}} \frac{\phi(k_r)}{k_r}
(z-k_rt_r)^{r-1}+ O\left( z^{r-1} \log z \right).$$ The sum appearing here is, applying (7), $$\sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} z^{r-1-j} t_r^j \sum_{k_rt_r\le z-t_1-\cdots
-t_{r-1} \atop{k_r\ge 1}} \phi(k_r)k_r^{j-1}$$ $$= \sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} z^{r-1-j} t_r^j \left( \frac{(z-t_1-\cdots
 -t_{r-1})^{j+1}}{(j+1)t_r^{j+1} \zeta(2)} +O(z^j \log z)\right)$$ $$=\frac1{t_r\zeta(2)} z^r \sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} \frac1{j+1} + O(z^{r-1}\log z)$$ $$=\frac1{r t_r \zeta(2)} z^r + O(z^{r-1}\log z),$$ which completes the proof.

Using estimate (4), $$\sum_{k_1t_1+\cdots +k_rt_r\le z \atop{j_1t_1+\cdots +j_rt_r\le z \atop{(k_1,j_1)=\cdots =(k_r,j_r)=1 \atop{k_1,j_1,...,k_r,j_r\ge 1}}}} 1
=
\sum_{k_1t_1+\cdots +k_rt_r\le z \atop{k_1,...,k_r\ge 1}}
\sum_{j_1t_1+\cdots +j_rt_r\le z \atop{(j_1,k_1)=\cdots =(j_r,k_r)=1 \atop{j_1,...,j_r\ge 1}}} 1$$ $$=\sum_{k_1t_1+\cdots +k_rt_r\le z \atop{k_1,...,k_r\ge 1}}
\left(\frac1{r!} \left( \prod_{i=1}^r \frac{\phi(k_i)}{k_i t_i} \right) z^r +
O\left(z^{r-1} \sum_{i=1}^r \theta(k_i) \right)\right)$$ $$=\frac{z^r}{r!\prod_{i=1}^r t_i} \sum_{k_1t_1+\cdots +k_rt_r\le z \atop{k_1,...,k_r\ge 1}}
\prod_{i=1}^r \frac{\phi(k_i)}{k_i} + O\left( z^{r-1} \sum_{i=1}^r \sum_{k_1t_1+\cdots +k_rt_r\le z} \theta(k_i) \right)$$ here the $O$-term is $O(z^{r-1}z^{r-1}z\log z)=O(z^{2r-1}\log z)$ and applying Lemma 2 to the main term finishes the proof.

Apply Lemma 3 for $z=\log x$, $t_1=\log p_1,...,t_r=\log p_r$.

This is a direct consequence of Theorem 2 and Corollary 1. The considered asymptotic density is $$\lim_{x\to \infty} P^{(e)}(p_1,...,p_r;x) (N(p_1,...,p_r;x))^{-2}=
%\qquad ( \sum_{p_1^{a_1}\cdots p_r^{a_r}\le x \atop{
%p_1^{b_1}\cdots p_r^{b_r}\le x \atop{
%(a_1,b_1)=...=(a_r,b_r)=1}}} 1 ) \cdot
%( \sum_{p_1^{a_1}\cdots p_r^{a_r}\le x} 1 )^{-2} =
(\zeta(2))^{-r}.$$

[99]{}

and [M. V. Subbarao]{}, The maximal order and the average order of multiplicative function $\sigma^{(e)}(n)$, [*Théorie des nombres. Proc. of the Int. Conf. Québec, 1987*]{}, de Gruyter, Berlin – New York, 1989, 201-206.

and [M. V. Subbarao]{}, On the distribution of exponential divisors, [*Annales Univ. Sci. Budapest., Sect. Comp.*]{}, [**22**]{} (2003), 161-180.

and [J. Wu]{}, On the sum of exponential divisors of an integer, [*Acta Math. Acad. Sci. Hung.*]{}, [**77**]{} (1997), 159-175.

, On an exponential totient function, [*Studia Univ. Babeş-Bolyai, Math.*]{}, [**41**]{} (1996), 91-94.

and [J. Wu]{}, On the exponential divisor function, [*Publ. Inst. Math. (Beograd) (N. S.)*]{}, [**61**]{} (1997), 21-32.

, On some arithmetic convolutions, in [*The Theory of Arithmetic Functions*]{}, Lecture Notes in Mathematics No. [**251**]{}, 247-271, Springer, 1972.

, [*Introduction to Analytic and Probabilistic Number Theory*]{}, Cambridge Univ. Press, 1995.

, Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, [*J. Théor. Nombres Bordeaux*]{}, [**7**]{} (1995), 133-141.

[[**László Tóth**]{}\
University of Pécs\
Institute of Mathematics and Informatics\
Ifjúság u. 6\
7624 Pécs, Hungary\
ltoth@ttk.pte.hu]{}
