GCD-closed sets and divisibility among power LCM matrices

Guangyan Zhu

doi:10.4153/S0008439526102276

GCD-closed sets and divisibility among power LCM matrices

Part of: Special matrices Polynomials and matrices Elementary number theory

Published online by Cambridge University Press: 09 July 2026

Guangyan Zhu

Show author details

Guangyan Zhu*: Affiliation:
School of Mathematics and Statistics, Hubei Minzu University , Enshi City, China
*: e-mail: 2009043@hbmzu.edu.cn

Article contents

Abstract
Introduction
Preliminary lemmas
A key lemma and proof of Theorem
Footnotes
References

Rights & Permissions

Abstract

Let $a,b,$ and n be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of n distinct positive integers. We denote by $[S^a]$ the $n\times n$ matrix having the ath power of the least common multiple of $x_i$ and $x_j$ as its $(i,j)$-entry. For $x\in S$, let ${G_{S}(x)=\{d\in S: d < x, d|x \ {\rm and} \ (d|y|x, y\in S)\Rightarrow y\in \{d,x\}\}}$. In this article, we show that $[S^a]$ divides $[S^b]$ in the ring of $n\times n$ matrices over the integers if $a|b$ and S is gcd closed (i.e., $\gcd(x_i, x_j)\in S$ for all integers i and j with $1\le i,j\le n$) and satisfies the condition $\mathcal G$ (That is, for any $x\in S$, either $G_S(x)$ contains at most one element, or $G_S(x)$ contains at least two elements and satisfies that ${\rm lcm}(y_1,y_2)=x$ and $\gcd(y_1,y_2)\in G_S(y_1)\cap G_S(y_2)$ for any $\{y_1,y_2\}\subseteq G_S(x)$). This confirms a conjecture of Hong proposed in (2026, Bull. Aust. Math. Soc., 113, 231–243).

Keywords

Divisibility power GCD matrix power LCM matrix greatest-type divisor gcd-closed set

MSC classification

Primary: 11C20: Matrices, determinants

Secondary: 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors 15B36: Matrices of integers

Information

Type: Article
Information: Canadian Mathematical Bulletin , First View , pp. 1 - 15

DOI: https://doi.org/10.4153/S0008439526102276 [Opens in a new window]
Copyright: © The Author(s), 2026. Published by Cambridge University Press on behalf of Canadian Mathematical Society

1 Introduction

Let $a,b,$ and n be positive integers. For any integers x and y, we denote by $(x, y)$ the greatest common divisor of integers x and y and by $[x,y]$ their least common multiple. Let $\mathbb Z$ denote the ring of integers and let $|T|$ stand for the cardinality of a finite set T of integers. Let f be an arithmetic function and let $S=\{x_1, \ldots , x_n \}$ . Let $(f(S))$ and $(f[S])$ denote the $n\times n$ matrices whose $(i,j)$ -entries are $f((x_i, x_j))$ and $f([x_i, x_j])$ , respectively. Define $\langle n\rangle :=\{1,\ldots , n\}$ . For any real number e, let $\xi _e$ be the arithmetic function defined for any positive integer x by $\xi _e(x)=x^e$ . The $n\times n$ matrices $(\xi _a(x_i, x_j)) =((x_i, x_j)^a)$ (abbreviated by $(S^a)$ ) and $(\xi _a[x_i, x_j])=([x_i, x_j]^a)$ (abbreviated by $[S^a]$ ) are called the ath power GCD matrix on S and the ath power LCM matrix on S, respectively. The set S is said to be factor closed (written as FC for short) if $(x\in S$ , $d>0$ , $d|x)\Rightarrow d\in S$ . We say that S is gcd closed if S contains $(x_i, x_j)$ for all integers i and j with $1\le i,j\le n$ . Obviously, any FC set contains 1 and is gcd closed but the converse is not true. For instance, if the set S is FC, then for any integer $x>1$ , the set $xS:=\{xy| y\in S\}$ is gcd closed but not FC since $1\not \in xS$ . One hundred and fifty years ago, Smith [Reference Smith26] proved that

(1.1) $$ \begin{align} \det([x_i,x_j])=\prod_{k=1}^n\varphi(x_k)\pi(x_k) \end{align} $$ det left parenthesis left bracket x Subscript i Baseline comma x Subscript j Baseline right bracket right parenthesis equals product Underscript k equals 1 Overscript n Endscripts phi left parenthesis x Subscript k Baseline right parenthesis pi left parenthesis x Subscript k Baseline right parenthesis

if S is FC, where $\varphi $ is the Euler’s totient function and $\pi $ is the multiplicative function defined for the prime power $p^r$ by $\pi (p^r)=-p$ . After that, many generalizations of Smith’s determinant (1.1) and related results were published (see, e.g., [Reference Apostol1–Reference McCarthy25] and [Reference Tan and Li27–Reference Zhu and Yu35]). In particular, an elegant result was gotten by Hong et al. [Reference Hong, Hu and Lin10] stating that for any integer $n\ge 2$ , one has

$$ \begin{align*}\det([i,j])_{2\le i,j\le n}=\Big(\prod_{k=1}^n \varphi (k)\pi(k)\Big)\sum_{t=1\atop t \ \text{is square free}}^n\frac{t\mu(t)}{\varphi (t)}, \end{align*} $$

where $\mu $ is the M $\ddot {\mathrm {o}}$ bius function and an integer $x\ge 1$ is called square free if x is not divisible by the square of any prime number.

Hong [Reference Hong12] introduced the concept of greatest-type divisor when he solved the famous Bourque–Ligh conjecture [Reference Bourque and Ligh4]. For any integer $x\in S$ , if $(y<x, y|z|x \ \mathrm {and}\ y,z\in S)\Rightarrow z\in \{y,x\},$ then y is called a greatest-type divisor of x. Let $G_S(x):=\{y\in S: y \ \text {is a greatest-type divisor of} \ x \ \mathrm {in} \ S\}$ . Let $M_n({\mathbb Z})$ stand for the ring of $n\times n$ matrices over the integers. In 1992, Bourque and Ligh [Reference Bourque and Ligh4] proved that $(S)$ divides $[S]$ in the ring $M_n(\mathbb Z)$ if S is FC. Namely, there exists a $B\in M_n({\mathbb Z})$ such that $[S]=B(S)$ or $[S]=(S)B$ . In 2002, Hong [Reference Hong14] showed that such a factorization is not true when S is gcd closed and $\max _{x \in S}\{|{G_S(x)}|\}=2$ .

On the other hand, Hong [Reference Hong17] initiated the investigation of divisibility among power LCM matrices. It was proved in [Reference Hong17] that $[S^a]|[S^b]$ if $a|b$ and S is a divisor chain (i.e., $x_{\sigma (1)}|\dots |x_{\sigma (n)}$ for a permutation $\sigma $ of $\langle n\rangle $ ). Evidently, a divisor chain is gcd closed but not conversely. Tan and Li [Reference Tan and Li27] generalized Hong’s result [Reference Hong17] by proving that $[S^a]|[S^b]$ holds in the ring $M_{|S|}({\mathbb Z})$ if $a|b$ and S consists of finitely many coprime divisor chains with $1\in S$ , and that such divisibility relation is not true if $a\nmid b$ . Zhu and Li [Reference Zhu and Li33] confirmed the conjecture of Hong raised in [Reference Hong17] stating that if $a|b$ and S is a gcd-closed set with $\max _{x\in S}\{|G_S(x)|\}=1$ , then $[S^a]\mid [S^b]$ in the ring $M_n({\mathbb Z})$ . In 2026, Hong [Reference Hong18] established the same divisibility result when S is FC. As in [Reference Hong18], for any set S of positive integers and for any $x\in S$ with $|G_S(x)|\ge 2$ , we say that the two distinct greatest-type divisors $y_1$ and $y_2$ of x in S satisfy the condition $\mathcal {G}$ if $[y_1,y_2]=x$ and $(y_1,y_2)\in G_S(y_1)\cap G_S(y_2)$ . We say that x satisfies the condition $\mathcal {G}$ if any two distinct greatest-type divisors of x in S satisfy the condition $\mathcal {G}$ . Moreover, we say that a set S of positive integers satisfies the condition $\mathcal {G}$ if any element x in S satisfies that either $|G_S(x)|\le 1$ , or $|G_S(x)|\ge 2$ and x satisfies the condition $\mathcal {G}$ . In [Reference Hong18], Hong showed that any FC set is a gcd-closed set satisfying the condition $\mathcal {G}$ . Furthermore, Hong raised the following conjecture in Section 3 of [Reference Hong18].

Conjecture 1.1 [Reference Hong18, Conjecture 3.4]

Let a and b be positive integers with $a|b$ and let S be a gcd-closed set satisfying the condition $\mathcal G$ . Then $[S^a]\mid [S^b]$ in the ring $M_{|S|}(\mathbb Z)$ .

For the case $\max _{x\in S}\{|G_S (x)|\}=1$ , by the theorem of Zhu and Li [Reference Zhu and Li33], we know that this conjecture is true. For the case $\max _{x\in S}\{|G_S (x)|\}=2$ , by [Reference Wan and Zhu28], one knows that Conjecture 1.1 holds. Furthermore, Zhu et al. [Reference Zhu, Luo and Wan34] confirmed Conjecture 1.1 for the case $\max _{x\in S}\{|G_S (x)|\}=3$ . But Conjecture 1.1 is still kept open when S is the general gcd-closed set satisfying the condition $\mathcal {G}$ .

In this article, our main goal is to study the divisibility among power LCM matrices. We introduce a new method to investigate Conjecture 1.1. The main result of this article can be stated as follows.

Theorem 1.2 Let S be a gcd-closed set satisfying the condition $\mathcal {G}$ and let a and b be positive integers such that $a|b$ . Then the ath power LCM matrix $[S^a]$ divides the bth power LCM matrix $[S^b]$ in the ring $M_{|S|}(\mathbb Z)$ .

This article is organized as follows. After introducing some necessary preliminary results, we proceed to the core of our argument. In fact, we first recall and show in Section 2 some results related to the arithmetic properties of gcd-closed sets that satisfy the condition $\mathcal G$ and then present a formula for the inverse of power LCM matrices. Finally, in Section 3, by proving a key lemma, we present the proof of Theorem 1.2.

The divisibility of matrices under consideration is invariant under relabeling the elements of S. Specifically, for any permutation $\sigma $ on $\langle n\rangle $ , we have $[S^a]\mid [S^b]$ if and only if $[S^a_{\sigma }]\mid [S^b_{\sigma }]$ . Therefore, without loss of generality, we may assume throughout the rest of this article that the set $S=\{x_1,\ldots ,x_n\}$ is ordered such that $x_1<\dots <x_n$ .

2 Preliminary lemmas

Throughout this section, let $x_m \in S$ with $G_S(x_m)=\{x_{m_1}, \ldots , x_{m_s}\}$ , $2\le m\le n=|S|$ and $1\le s\le m-1$ . Let $1\le k\le s$ and for any integers $i_1,\ldots , i_k$ with $1 \le i_1<\cdots <i_k \le s$ , we define

$$ \begin{align*}x_{m_{i_1\dots i_k}}:=(x_{m_{i_1}},\ldots, x_{m_{i_k}}).\end{align*} $$

Then $x_{m_{1\dots s}}=(x_{m_1}, \ldots , x_{m_s})$ . In what follows, we recall and show several results that are needed in the proof of our main result. For any $j\in \langle s\rangle $ , define $P_j:=\frac {x_m}{x_{m_j}}$ . We begin with a known result which is due to Zhu and Yu [Reference Zhu and Yu35] and is about the arithmetic property of the gcd-closed set S which satisfies the condition $\mathcal {G}$ .

Lemma 2.1 [Reference Zhu and Yu35]

Let S be a gcd-closed set satisfying the condition $\mathcal {G}$ and $x_m \in S$ with $G_S(x_m)=\{x_{m_1}, \ldots , x_{m_s}\}$ . Let k be an integer with $1 \le k \le s$ . Then for any integers $i_1,\ldots ,i_k$ with $1\le i_1<\cdots <i_k\le s$ , we have

$$ \begin{align*} x_{m_{i_1\dots i_k}} P_{i_1}\dots P_{i_k}=x_m. \end{align*} $$

Lemma 2.2 [Reference Zhu32]

Let S be a gcd-closed set satisfying the condition $\mathcal {G}$ and $x_l, x_m\in S$ with $G_S(x_m)=\{x_{m_1},\ldots ,x_{m_s}\}$ . Let $1\le k\le s-1$ . Assume that $(x_l,x_m)\mid x_{m_{1\dots k}}$ and $(x_l,x_m) \nmid x_{m_j}$ for all integers j with $k+1 \le j \le s$ . Let $1 \le i_1<\cdots < i_h \le k$ with $1\le h\le k$ and $k+1 \le j_1<\cdots <j_t \le s$ with $1\le t\le s-k$ . Then each of the following is true:

(i). $(x_l, x_{m_{i_1\dots i_h}})=(x_l, x_m).$

(ii). $[x_l,x_{m_{j_1\dots j_t}}]=[x_l,x_m].$

(iii). $[x_{l}, x_{m_{i_1\dots i_hj_1\dots j_t}}]=[x_l,x_{m_{i_1\dots i_h}}].$

Lemma 2.3 [Reference Bourque and Ligh6, Theorem 3]

If S is gcd-closed set of n positive integers and $(f(S))$ is nonsingular, then

$$ \begin{align*}((f(S))^{-1})_{ij}:={\underset{x_i|x_k\atop x_j|x_k} {\sum}}\frac{c_{ik}c_{jk}}{\alpha_f(x_k)}\end{align*} $$

with

(2.1) $$ \begin{align} \alpha_f(x_k):={\underset{d|x_k\atop d\nmid x_t, x_t<x_k}{\sum}}(f*\mu)(d) \end{align} $$

and

(2.2) $$ \begin{align} c_{ij}:=\sum _{dx_i|x_j\atop dx_i\nmid x_t, x_t<x_j}\mu(d). \end{align} $$ c Subscript i j Baseline colon equals sigma summation Underscript StartLayout 1st Row d x Subscript i Baseline vertical bar x Subscript j Baseline 2nd Row d x Subscript i Baseline does not divide x Subscript t Baseline comma x Subscript t Baseline less than x Subscript j Baseline EndLayout Endscripts mu left parenthesis d right parenthesis period

Lemma 2.4 [Reference Hong16]

Let S be a gcd-closed set and let $\alpha _{f}(x_k)$ be given as in (2.1). Then

$$ \begin{align*}\alpha_{f}(x_k)=\sum\limits_{J\subseteq G_S(x_k)}(-1)^{|J|} f\big(\gcd (J\cup \{x_k\})\big). \end{align*} $$

Lemma 2.5 [Reference Bourque and Ligh6]

If S is gcd closed, then

$$ \begin{align*} \det(f(S))=\prod\limits_{k=1}^n\alpha_{f}(x_k) \end{align*} $$ StartLayout 1st Row det left parenthesis f left parenthesis upper S right parenthesis right parenthesis equals product Underscript k equals 1 Overscript n Endscripts alpha Subscript f Baseline left parenthesis x Subscript k Baseline right parenthesis EndLayout

with $\alpha _{f}(x_k)$ being given as in (2.1).

Lemma 2.6 [Reference Zhu and Yu35]

Let S be a gcd-closed set satisfying the condition $\mathcal G$ and let $x_m\in S$ be such that $\{x_{m_1},\ldots ,x_{m_s}\}\subseteq G_S(x_m)$ . For any real number e, define

$$ \begin{align*}\beta_{\xi_e}(s,x_m) :=x_m^e+\sum\limits_{t=1}^s(-1)^t\sum \limits_{1\le i_1<\cdots<i_t\le s}x^e_{m_{i_1\dots i_t}}. \end{align*} $$

Then

$$ \begin{align*} \beta_{\xi_e}(s,x_m)=x^e_{m_{1\dots s}} \prod_{j=1}^s\Big(\Big(\frac{x_m}{x_{m_j}}\Big)^e-1\Big). \end{align*} $$

For any real number e, the reciprocal arithmetic function $\frac {1}{\xi _e}$ is defined for any positive integer x by $\frac {1}{\xi _e}(x):=\frac {1}{x^e}$ . In fact, one has $\frac {1}{\xi _e}=\xi _{-e}$ .

Lemma 2.7 Let S be a gcd-closed set satisfying the condition $\mathcal G$ and $x_m\in S$ with $G_S(x_m)=\{x_{m_1},\ldots ,x_{m_s}\}$ . Then

$$ \begin{align*}\alpha_{\frac{1}{\xi_a}}(x_m) =x_{m_{1\dots s}}^{-a} \prod_{j=1}^s\Big(\Big(\frac{x_m}{x_{m_j}}\Big)^{-a}-1\Big). \end{align*} $$

Proof Setting $f=\frac {1}{\xi _a}$ in Lemma 2.4, by Lemmas 2.4 and 2.6, we get that

$$ \begin{align*} \alpha_{\frac{1}{\xi_a}}(x_m)=\beta_{\xi_{-a}}(s,x_m) =x^{-a}_{m_{1\dots s}}\prod_{j=1}^s\Big(\Big(\frac{x_m}{x_{m_j}}\Big)^{-a}-1\Big) \end{align*} $$

as required. So Lemma 2.7 is proved.

For any real number e and for any gcd-closed set S satisfying the condition $\mathcal {G}$ , Zhu and Yu [Reference Zhu and Yu35] proved that the eth power LCM matrix $[S^e]$ is nonsingular when $e\ne 0$ which confirmed another conjecture of Hong raised in [Reference Hong18]. In the following, we present the explicit formulas for the inverses of the eth power GCD matrix $(S^e)$ and the eth power LCM matrix $[S^e]$ .

Lemma 2.8 Let S be a gcd-closed set of n positive integers and let S satisfy the condition $\mathcal G$ . Then for any nonzero real number e, the eth power GCD matrix $(S^e)$ and the eth power LCM matrix $[S^e]$ are both nonsingular, and for any integers i and j with $1\le i,j\le n$ , we have

$$ \begin{align*}((S^e)^{-1})_{ij}:={\underset{x_i|x_k\atop x_j|x_k}{\sum}} \frac{c_{ik}c_{jk}}{\alpha_{\xi_e}(x_k)} \end{align*} $$

and

$$ \begin{align*}([S^e]^{-1})_{ij}:=\frac{1}{x_i^e x_j^e}{\underset{x_i|x_k\atop x_j|x_k}{\sum}} \frac{c_{ik}c_{jk}}{\alpha_{\frac{1}{\xi_e}}(x_k)}, \end{align*} $$

left parenthesis left bracket upper S Superscript e Baseline right bracket Superscript negative 1 Baseline right parenthesis Subscript i j Baseline colon equals StartFraction 1 Over x Subscript i Superscript e Baseline x Subscript j Superscript e Baseline EndFraction sigma summation Underscript StartLayout 1st Row x Subscript i Baseline vertical bar x Subscript k Baseline 2nd Row x Subscript j Baseline vertical bar x Subscript k Baseline EndLayout Endscripts StartFraction c Subscript i k Baseline c Subscript j k Baseline Over alpha Subscript StartFraction 1 Over xi Sub Subscript e Subscript EndFraction Baseline left parenthesis x Subscript k Baseline right parenthesis EndFraction comma

where $c_{ik}$ and $c_{jk}$ are defined as in (2.2).

Proof Since

$$ \begin{align*}{[x_i,x_j]}^e=\frac{x_i^e x_j^e}{{(x_i,x_j)}^e},\end{align*} $$ left bracket x Subscript i Baseline comma x Subscript j Baseline right bracket Superscript e Baseline equals StartFraction x Subscript i Superscript e Baseline x Subscript j Superscript e Baseline Over left parenthesis x Subscript i Baseline comma x Subscript j Baseline right parenthesis Superscript e Baseline EndFraction comma

we have

(2.3) $$ \begin{align} [S^e] &=\mathrm{diag}(x_1^e,\ldots,x_n^e)\cdot \Big(\frac{1}{\xi_e}(x_i,x_j)\Big)\cdot\mathrm{diag}(x_1^e,\ldots,x_n^e)\notag\\ &=:\mathrm{diag}(x_1^e,\ldots,x_n^e)\cdot B\cdot \mathrm{diag}(x_1^e,\ldots,x_n^e), \end{align} $$

StartLayout 1st Row 1st Column left bracket upper S Superscript e Baseline right bracket 2nd Column equals diag left parenthesis x 1 Superscript e Baseline comma ellipsis comma x Subscript n Superscript e Baseline right parenthesis dot left parenthesis StartFraction 1 Over xi Subscript e Baseline EndFraction left parenthesis x Subscript i Baseline comma x Subscript j Baseline right parenthesis right parenthesis dot diag left parenthesis x 1 Superscript e Baseline comma ellipsis comma x Subscript n Superscript e Baseline right parenthesis 2nd Row 1st Column Blank 2nd Column equals colon diag left parenthesis x 1 Superscript e Baseline comma ellipsis comma x Subscript n Superscript e Baseline right parenthesis dot upper B dot diag left parenthesis x 1 Superscript e Baseline comma ellipsis comma x Subscript n Superscript e Baseline right parenthesis comma EndLayout

where $B:=\big (\frac {1}{\xi _e}(x_i,x_j)\big )=(\xi _{-e}(S))$ is the $(-e)$ th power GCD matrix defined on S. We take the determinants on both sides of (2.3) and obtain that

(2.4) $$ \begin{align} \det[S^e]=\det B\cdot\prod\limits_{k=1}^nx_k^{2e}. \end{align} $$ det left bracket upper S Superscript e Baseline right bracket equals det upper B dot product Underscript k equals 1 Overscript n Endscripts x Subscript k Superscript 2 e Baseline period

Since the two diagonal matrices in (2.3) are nonsingular, the nonsingularity of the eth power LCM matrix $[S^e]$ is equivalent to that of B. Thus, we first prove that B is nonsingular.

By Lemma 2.5, the determinant of B is equal to

(2.5) $$ \begin{align} \det B=\prod\limits_{k=1}^n\alpha_{\frac{1}{\xi_e}}(x_k). \end{align} $$ det upper B equals product Underscript k equals 1 Overscript n Endscripts alpha Subscript StartFraction 1 Over xi Sub Subscript e Subscript EndFraction Baseline left parenthesis x Subscript k Baseline right parenthesis period

Hence, it suffices to show that each factor $\alpha _{\frac {1}{\xi _e}}(x_k)$ is nonzero. First, from Lemma 2.4, we have

(2.6) $$ \begin{align} \alpha_{\frac{1}{\xi_e}}(x_1)=x_1^{-e}\ne0. \end{align} $$

Next, for any $2\le m\le n=|S|$ , let $G_S(x_m)=\{x_{m_1},\ldots , x_{m_s}\}$ with $s=|G_S(x_m)|\ge 1$ . For each $x_{m_j}\in G_S(x_m)$ ( $1\le j\le s$ ), since $-e\ne 0$ and $x_m\ne x_{m_j}$ , we have

$$ \begin{align*}\Big(\frac{x_m}{x_{m_j}}\Big)^{-e}-1\neq0. \end{align*} $$ left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript negative e Baseline minus 1 not equals 0 period

Applying Lemma 2.7 then yields that

(2.7) $$ \begin{align} \alpha_{\frac{1}{\xi_e}}(x_m) =x_{m_{1\dots s}}^{-e} \prod_{j=1}^s\Big(\Big(\frac{x_m}{x_{m_j}}\Big)^{-e}-1\Big)\ne 0. \end{align} $$

Combining (2.5)–(2.7), we conclude that $\det B\ne 0$ , so B is nonsingular. Replacing $-e$ by e, this gives that the eth power GCD matrix $(S^e)$ is nonsingular since $e\ne 0$ . Moreover, together with (2.4), this implies that $[S^e]$ is nonsingular.

Furthermore, we apply Lemma 2.3 to the function $f=\xi _e$ and get the formula for $(S^e)^{-1}$ . Letting $f=\frac {1}{\xi _e}$ in Lemma 2.3 gives the entrywise formula for $B^{-1}$ as follows:

(2.8) $$ \begin{align} \big(B^{-1}\big)_{ij} ={\underset{x_i|x_k\atop x_j|x_k} {\sum}}\frac{c_{i k}c_{j k}}{\alpha_{\frac{1}{\xi_e}}(x_k)}. \end{align} $$

Finally, combining (2.3) and (2.8) yields the desired result immediately.

This completes the proof of Lemma 2.8.

Lemma 2.9 [Reference Zhu32]

Let S be a gcd-closed set satisfying the condition $\mathcal G$ with $x_m \in S$ . For any integer r with $1\le r\le m$ , if $c_{rm}$ is defined as in (2.2) and $G_S(x_m)=\{x_{m_1},\ldots , x_{m_s}\}$ , then

$$ \begin{align*} c_{rm}=\left\{\begin{array}{@{}cl} 1, &\text{if}\ r=m,\\ (-1)^k, &\text{if}\ r=m_{i_1\dots i_k}\ (1 \leq i_1 < \cdots < i_k \leq s,\ 1 \leq k \leq s),\\ 0, &\text{otherwise}. \end{array}\right. \end{align*} $$

c Subscript r m Baseline equals StartLayout Enlarged left brace 1st Row 1st Column 1 comma 2nd Column if r equals m comma 2nd Row 1st Column left parenthesis negative 1 right parenthesis Superscript k Baseline comma 2nd Column if r equals m Subscript i 1 ellipsis i Sub Subscript k Subscript Baseline left parenthesis 1 less than or equals i 1 less than midline horizontal ellipsis less than i Subscript k Baseline less than or equals s comma 1 less than or equals k less than or equals s right parenthesis comma 3rd Row 1st Column 0 comma 2nd Column otherwise period EndLayout

3 A key lemma and proof of Theorem 1.2

In this section, with the aid of the lemmas presented in Section 2, we prove Theorem 1.2. For any integers l and m with $1\le l,m\le |S|$ , we define the two-variable function H as follows:

(3.1) $$ \begin{align} H(l,m):=\frac{1}{\alpha_{\frac{1}{\xi_a}}(x_m)}\sum\limits_{x_r|x_m}\frac{c_{rm}[x_l, x_r]^b}{x_r^a}. \end{align} $$

First of all, we prove a key lemma that plays a crucial role in the proof of Theorem 1.2.

Lemma 3.1 Let S be a gcd-closed set satisfying the condition $\mathcal {G}$ with $x_l,x_m\in S$ , and let $G_S(x_m)=\{x_{m_1},\ldots , x_{m_s}\}$ with $s\ge 1$ . Let a and b be positive integers. Then

$$ \begin{align*} H(l,m)=\left\{ \begin{aligned} & (-1)^s \Big(\frac{x_l}{(x_l,x_m)}\Big)^b x_m^a x_{m_{1\dots s}}^{b-a} \prod_{j=1}^s\frac{\big(\frac{x_m}{x_{m_j}}\big)^{b-a}-1}{\big(\frac{x_m}{x_{m_j}}\big)^a-1}, \ \ \ \text{if}\ (x_l,x_m)\mid x_{m_{1\dots s}},\\ & x_l^b, \ \ \ \text{if}\ x_m\mid x_l,\\ & (-1)^k \Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_m^ax_{m_{1\dots k}}^{b-a} \prod_{j=1}^k\frac{\big(\frac{x_m}{x_{m_j}}\big)^{b-a}-1}{\big(\frac{x_m}{x_{m_j}}\big)^a-1},\ \ \ \text{if}\ (x_l,x_m)\mid x_{m_{1\dots k}}\\ &\text{for some}\ 1\le k\le s-1\ \text{and}\ (x_l,x_m) \nmid x_{m_j}\ \text{for all}\ k+1 \le j \le s. \end{aligned} \right. \end{align*} $$

upper H left parenthesis l comma m right parenthesis equals StartLayout Enlarged left brace 1st Row 1st Column Blank 2nd Column left parenthesis negative 1 right parenthesis Superscript s Baseline left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Superscript a Baseline x Subscript m Sub Subscript 1 ellipsis s Subscript Superscript b minus a Baseline product Underscript j equals 1 Overscript s Endscripts StartFraction left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 Over left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript a Baseline minus 1 EndFraction comma if left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis vertical bar x Subscript m Sub Subscript 1 ellipsis s Subscript Baseline comma 2nd Row 1st Column Blank 2nd Column x Subscript l Superscript b Baseline comma if x Subscript m Baseline vertical bar x Subscript l Baseline comma 3rd Row 1st Column Blank 2nd Column left parenthesis negative 1 right parenthesis Superscript k Baseline left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Superscript a Baseline x Subscript m Sub Subscript 1 ellipsis k Subscript Superscript b minus a Baseline product Underscript j equals 1 Overscript k Endscripts StartFraction left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 Over left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript a Baseline minus 1 EndFraction comma if left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis vertical bar x Subscript m Sub Subscript 1 ellipsis k Subscript Baseline 4th Row 1st Column Blank 2nd Column for some 1 less than or equals k less than or equals s minus 1 and left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis does not divide x Subscript m Sub Subscript j Subscript Baseline for all k plus 1 less than or equals j less than or equals s period EndLayout

Proof By (3.1) and Lemma 2.9, we have

(3.2) $$ \begin{align} \alpha_{\frac{1}{\xi_a}}(x_m)H(l,m)&= \sum\limits_{x_r|x_m}\frac{c_{rm}[x_l, x_r]^b}{x_r^a}\notag\\ &=\frac{[x_l,x_m]^b}{x_m^a}+\sum\limits_{t=1}^s(-1)^t\sum\limits_{1\le j_{1}<\cdots<j_{t}\le s} \frac{[x_l,x_{m_{j_{1}\dots j_{t}}}]^b}{x_{m_{j_{1}\dots j_{t}}}^a}. \end{align} $$

StartLayout 1st Row 1st Column alpha Subscript StartFraction 1 Over xi Sub Subscript a Subscript EndFraction Baseline left parenthesis x Subscript m Baseline right parenthesis upper H left parenthesis l comma m right parenthesis 2nd Column equals sigma summation Underscript x Subscript r Baseline vertical bar x Subscript m Baseline Endscripts StartFraction c Subscript r m Baseline left bracket x Subscript l Baseline comma x Subscript r Baseline right bracket Superscript b Baseline Over x Subscript r Superscript a Baseline EndFraction 2nd Row 1st Column Blank 2nd Column equals StartFraction left bracket x Subscript l Baseline comma x Subscript m Baseline right bracket Superscript b Baseline Over x Subscript m Superscript a Baseline EndFraction plus sigma summation Underscript t equals 1 Overscript s Endscripts left parenthesis negative 1 right parenthesis Superscript t Baseline sigma summation Underscript 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s Endscripts ContinuedFraction left bracket x Subscript l Baseline comma x Subscript m Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Baseline right bracket Superscript b Baseline Over x Subscript m Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Superscript a Baseline period EndLayout

Since $(x_l,x_m)|x_m$ , it follows that either $(x_l,x_m)|x_{m_{1\dots s}}$ , or $(x_l,x_m)=x_m $ , or there exist an integer k with $1\le k\le s-1$ and k integers $i_1,\ldots , i_k$ with $1\le i_1<\cdots <i_k\le s$ such that $(x_l,x_m)|x_{m_{i_1\dots i_k}}$ and $(x_l,x_m) \nmid x_{m_j} $ for all $j \in \langle s\rangle \setminus \{ i_{1 },\ldots , i_{k}\}$ . So we only need to consider the following three cases.

Case 1. $(x_l,x_m)|x_{m_{1\dots s}}$ . For $1 \le j_1<\cdots <j_t\le s$ with $1\le t\le s$ , since $x_{m_{1\dots s}}| x_{m_{j_1\dots j_t}}|x_m$ , we have $(x_l,x_m)|x_{m_{j_1\dots j_t}}|x_m$ and

$$ \begin{align*} (x_l,x_{m_{j_1\dots j_t}}) = (x_l,(x_{m_{j_1\dots j_t}}, x_m)) =((x_l,x_m),x_{m_{j_1\dots j_t}}) =(x_l,x_m). \end{align*} $$

Thus, by (3.2) and Lemma 2.6, we have

$$ \begin{align*} \alpha_{\frac{1}{\xi_a}}(x_m)H(l,m)&=\dfrac{x_l^bx_m^b}{x_m^a(x_l,x_m)^b}+\sum\limits_{t=1}^s(-1)^t \sum\limits_{1\le j_{1}<\cdots<j_{t}\le s}\dfrac{x_l^bx^b_{m_{j_1\dots j_t}}}{x^a_{m_{j_1\dots j_t}}(x_l,x_{m_{j_1\dots j_t}})^b}\notag\\ &=\dfrac{x_l^bx_m^{b-a}}{(x_l,x_m)^b} +\sum\limits_{t=1}^s(-1)^t \sum\limits_{1\le j_{1}<\cdots<j_{t}\le s}\dfrac{x_l^bx^b_{m_{j_1\dots j_t}}}{x^a_{m_{j_1\dots j_t}}(x_l,x_m)^b}\notag\\ &=\dfrac{x_l^bx_m^{b-a}}{(x_l,x_m)^b}+\dfrac{x_l^b}{(x_l,x_m)^b}\sum\limits_{t=1}^s(-1)^t \sum\limits_{1\le j_{1}<\cdots<j_{t}\le s}\frac{x^b_{m_{j_1\dots j_t}}}{x^a_{m_{j_1\dots j_t}}}\notag\\ &=\dfrac{x_l^b}{(x_l,x_m)^b}\Big(x_m^{b-a}+\sum\limits_{t=1}^s(-1)^t \sum\limits_{1\le j_{1}<\cdots<j_{t}\le s}x_{m_{j_1\dots j_t}}^{b-a}\Big)\notag\\ &=\dfrac{x_l^b}{(x_l,x_m)^b}\beta_{\xi_{b-a}}(s,x_m)\notag\\ &=\dfrac{x_l^b}{(x_l,x_m)^b}x_{m_{1\dots s}}^{b-a}\prod_{j=1}^s\big(\Big(\frac{x_m}{x_{m_j}}\Big)^{b-a}-1\big).\notag \end{align*} $$

Then it follows from Lemmas 2.1 and 2.7 that

$$ \begin{align*} H(l,m)&=\Big(\frac{x_l}{(x_l,x_m)}\Big)^b\frac{x_{m_{1\dots s}}^{b-a}\prod_{j=1}^s\big(\big(\frac{x_m}{x_{m_j}}\big)^{b-a}-1\big)} {x_{m_{1\dots s}}^{-a}\prod_{j=1}^s\big(\big(\frac{x_m}{x_{m_j}}\big)^{-a}-1\big)}\\ &=\Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_{m_{1\dots s}}^b(-1)^s\prod_{j=1}^s\Big(\frac{x_m}{x_{m_j}}\Big)^a\cdot \prod_{j=1}^s \frac{\big(\frac{x_m}{x_{m_j}}\big)^{b-a}-1}{\big(\frac{x_m}{x_{m_j}}\big)^a-1}\\ &=\Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_{m_{1\dots s}}^b(-1)^s\Big(\frac{x_m}{x_{m_{1\dots s}}}\Big)^a \prod_{j=1}^s \frac{\big(\frac{x_m}{x_{m_j}}\big)^{b-a}-1}{\big(\frac{x_m}{x_{m_j}}\big)^a-1}\\ &=\Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_{m_{1\dots s}}^{b-a}x_m^a (-1)^s \prod_{j=1}^s \frac{\big(\frac{x_m}{x_{m_j}}\big)^{b-a}-1}{\big(\frac{x_m}{x_{m_j}}\big)^a-1}\\ &=(-1)^s \Big(\frac{x_l}{(x_l,x_m)}\Big)^b x_m^a x_{m_{1\dots s}}^{b-a} \prod_{j=1}^s \frac{\big(\frac{x_m}{x_{m_j}}\big)^{b-a}-1}{\big(\frac{x_m}{x_{m_j}}\big)^a-1} \end{align*} $$

StartLayout 1st Row 1st Column upper H left parenthesis l comma m right parenthesis 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline ContinuedFraction x Subscript m Sub Subscript 1 ellipsis s Subscript Superscript b minus a Baseline product Underscript j equals 1 Overscript s Endscripts left parenthesis left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 right parenthesis Over x Subscript m Sub Subscript 1 ellipsis s Subscript Superscript negative a Baseline product Underscript j equals 1 Overscript s Endscripts left parenthesis left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript negative a Baseline minus 1 right parenthesis 2nd Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Sub Subscript 1 ellipsis s Superscript b Baseline left parenthesis negative 1 right parenthesis Superscript s Baseline product Underscript j equals 1 Overscript s Endscripts left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript a Baseline dot product Underscript j equals 1 Overscript s Endscripts StartFraction left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 Over left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript a Baseline minus 1 EndFraction 3rd Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Sub Subscript 1 ellipsis s Superscript b Baseline left parenthesis negative 1 right parenthesis Superscript s Baseline left parenthesis ContinuedFraction x Subscript m Baseline Over x Subscript m Sub Subscript 1 ellipsis s Subscript Baseline right parenthesis Superscript a Baseline product Underscript j equals 1 Overscript s Endscripts StartFraction left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 Over left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript a Baseline minus 1 EndFraction 4th Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Sub Subscript 1 ellipsis s Superscript b minus a Baseline x Subscript m Superscript a Baseline left parenthesis negative 1 right parenthesis Superscript s Baseline product Underscript j equals 1 Overscript s Endscripts StartFraction left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 Over left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript a Baseline minus 1 EndFraction 5th Row 1st Column Blank 2nd Column equals left parenthesis negative 1 right parenthesis Superscript s Baseline left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Superscript a Baseline x Subscript m Sub Subscript 1 ellipsis s Superscript b minus a Baseline product Underscript j equals 1 Overscript s Endscripts StartFraction left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 Over left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript a Baseline minus 1 EndFraction EndLayout

as expected.

Case 2. $(x_l,x_m)=x_m$ . Then $x_m|x_l$ . Clearly, $x_{m_{j_1\dots j_t}}|x_m|x_l$ for $1 \le j_1<\cdots <j_t\le s$ . This implies that

$$ \begin{align*}[x_l,x_m]=[x_l, x_{m_{j_1j_2\dots j_t}}]=x_l.\end{align*} $$

Therefore, by (3.2) and Lemmas 2.6 and 2.7, we have

$$ \begin{align*} \alpha_{\frac{1}{\xi_a}}(x_m)H(l,m)&=\frac{x_l^b}{x_m^a}+\sum\limits_{t=1}^s(-1)^t\sum\limits_{1\le j_{1}<\cdots<j_{t}\le s} \frac{[x_l, x_{m_{j_1\dots j_t}}]^b}{x_{m_{j_1\dots j_t}}^a}\\ &=\frac{x_l^b}{x_m^a}+\sum\limits_{t=1}^s(-1)^t\sum\limits_{1\le j_{1}<\cdots<j_{t}\le s}\frac{x_l^b}{x_{m_{j_1\dots j_t}}^a}\\ &=x_l^b\Big(x_m^{-a}+\sum\limits_{t=1}^s(-1)^t\sum\limits_{1\le j_{1}<\cdots<j_{t}\le s}x_{m_{j_1\dots j_t}}^{-a}\Big)\\ &=x_l^b\beta_{\xi_{-a}}(s,x_m) = x_l^b\alpha_{\frac{1}{\xi_a}}(x_m), \end{align*} $$

and so $H(l,m)=x_l^b$ as desired.

Case 3. There is an integer k with $1\le k\le s-1$ and k integers $i_1,\ldots , i_k$ with $1\le i_1<\cdots <i_k\le s-1$ such that $(x_l,x_m)|x_{m_{i_1\dots i_k}}$ and $(x_l,x_m) \nmid x_{m_j} $ for all $j \in \langle s\rangle \setminus \{i_{1}, \ldots , i_{k}\}$ . WLOG, one may let $(x_l,x_m)\mid x_{m_{1\dots k}}$ and $(x_l,x_m) \nmid x_{m_j} $ for all $k+1 \le j \le s$ . By (3.2), we get that

$$ \begin{align*} \alpha_{\frac{1}{\xi_a}}(x_m)H(l,m)=\ &\frac{[x_l,x_m]^b}{x_m^a}+\sum_{h=1}^{k}(-1)^{h}\sum\limits_{1 \le i_1<\cdots<i_h\le k} \frac{[x_l,x_{m_{i_1\dots i_h}}]^b}{x_{m_{i_1\dots i_h}}^a}\\ &+\sum_{t=1}^{s-k}(-1)^{t}\sum\limits_{k+1 \le j_1<\cdots <j_t \le s}\frac{[x_l,x_{m_{j_1\dots j_t}}]^b}{x_{m_{j_1\dots j_t}}^a} \\ &+\sum_{h=1}^{k}\sum_{t=1}^{s-k}(-1)^{h+t}\sum\limits_{1 \le i_1<\cdots<i_h\le k \atop k+1 \le j_1<\cdots <j_t \le s} \frac{[x_l,x_{m_{i_1\dots i_hj_1\dots j_t}}]^b}{x_{m_{i_1\dots i_hj_1\dots j_t}}^a}\\ =\ &E_1+E_2, \end{align*} $$

where

$$ \begin{align*} E_1:=\frac{[x_l,x_m]^b}{x_m^a} +\sum_{h=1}^{k}(-1)^{h}\sum\limits_{1 \le i_1<\cdots<i_h\le k}\frac{[x_l,x_{m_{i_1\dots i_h}}]^b}{x_{m_{i_1\dots i_h}}^a} \end{align*} $$

and

$$ \begin{align*} E_2:=&\sum_{t=1}^{s-k}(-1)^{t}\sum\limits_{k+1 \le j_1<\cdots <j_t \le s}\frac{[x_l,x_{m_{j_1\dots j_t}}]^b}{x_{m_{j_1\dots j_t}}^a}\nonumber\\ &+\sum_{h=1}^{k}\sum_{t=1}^{s-k}(-1)^{h+t}\sum\limits_{1 \le i_1<\cdots<i_h\le k \atop k+1 \le j_1<\cdots <j_t \le s} \frac{[x_l,x_{m_{i_1\dots i_hj_1\dots j_t}}]^b}{x_{m_{i_1\dots i_hj_1\dots j_t}}^a}. \end{align*} $$

StartLayout 1st Row 1st Column upper E 2 colon equals 2nd Column sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript t Baseline sigma summation Underscript k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s Endscripts ContinuedFraction left bracket x Subscript l Baseline comma x Subscript m Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Baseline right bracket Superscript b Baseline Over x Subscript m Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Superscript a Baseline 2nd Row 1st Column Blank 2nd Column plus sigma summation Underscript h equals 1 Overscript k Endscripts sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript h plus t Baseline sigma summation Underscript StartLayout 1st Row 1 less than or equals i 1 less than midline horizontal ellipsis less than i Subscript h Baseline less than or equals k 2nd Row k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s EndLayout Endscripts ContinuedFraction left bracket x Subscript l Baseline comma x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Baseline right bracket Superscript b Baseline Over x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Superscript a Baseline period EndLayout

First of all, we compute $E_1$ . By Lemmas 2.2(i) and 2.6, we obtain that

$$ \begin{align*} E_1&=\frac{[x_l,x_m]^b}{x^a_m}+\sum_{h=1}^{k}(-1)^{h}\sum\limits_{1 \le i_1<\cdots<i_h\le k} \frac{[x_l,x_{m_{i_1\dots i_h}}]^b}{x_{m_{i_1\dots i_h}}^a} \\ &=\frac{x_l^bx_m^b}{x^a_m(x_l,x_m)^b}+\sum_{h=1}^{k}(-1)^{h}\sum\limits_{1 \le i_1<\cdots<i_h\le k} \frac{x_l^bx_{m_{i_1\dots i_h}}^b}{x_{m_{i_1\dots i_h}}^a(x_l,x_{m_{i_1\dots i_h}})^b} \\ &=\frac{x_l^bx_m^{b-a}}{(x_l,x_m)^b}+\sum_{h=1}^{k}(-1)^{h}\sum\limits_{1\le i_1<\cdots<i_h\le k} \frac{x_l^bx_{m_{i_1\dots i_h}}^{b-a}}{(x_l,x_m)^b} \\ &=\Big(\frac{x_l}{(x_l,x_m)}\Big)^b\Big(x_m^{b-a}+\sum_{h=1}^{k}(-1)^{h} \sum\limits_{1\le i_1<\cdots<i_h\le k}x_{m_{i_1\dots i_h}}^{b-a}\Big) \\ &=\Big(\frac{x_l}{(x_l,x_m)}\Big)^b\beta_{\xi_{b-a}}(k,x_m) \\ &=\Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_{m_{1\dots k}}^{b-a}\prod_{j=1}^k\Big(\Big(\frac{x_m}{x_{m_j}}\Big)^{b-a}-1\Big). \end{align*} $$

StartLayout 1st Row 1st Column upper E 1 2nd Column equals StartFraction left bracket x Subscript l Baseline comma x Subscript m Baseline right bracket Superscript b Baseline Over x Subscript m Superscript a Baseline EndFraction plus sigma summation Underscript h equals 1 Overscript k Endscripts left parenthesis negative 1 right parenthesis Superscript h Baseline sigma summation Underscript 1 less than or equals i 1 less than midline horizontal ellipsis less than i Subscript h Baseline less than or equals k Endscripts ContinuedFraction left bracket x Subscript l Baseline comma x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript Subscript Baseline right bracket Superscript b Baseline Over x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript Subscript Superscript a Baseline 2nd Row 1st Column Blank 2nd Column equals StartFraction x Subscript l Superscript b Baseline x Subscript m Superscript b Baseline Over x Subscript m Superscript a Baseline left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis Superscript b Baseline EndFraction plus sigma summation Underscript h equals 1 Overscript k Endscripts left parenthesis negative 1 right parenthesis Superscript h Baseline sigma summation Underscript 1 less than or equals i 1 less than midline horizontal ellipsis less than i Subscript h Baseline less than or equals k Endscripts ContinuedFraction x Subscript l Superscript b Baseline x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript Subscript Superscript b Baseline Over x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript Subscript Superscript a Baseline left parenthesis x Subscript l Baseline comma x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript Subscript Baseline right parenthesis Superscript b Baseline 3rd Row 1st Column Blank 2nd Column equals StartFraction x Subscript l Superscript b Baseline x Subscript m Superscript b minus a Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis Superscript b Baseline EndFraction plus sigma summation Underscript h equals 1 Overscript k Endscripts left parenthesis negative 1 right parenthesis Superscript h Baseline sigma summation Underscript 1 less than or equals i 1 less than midline horizontal ellipsis less than i Subscript h Baseline less than or equals k Endscripts StartFraction x Subscript l Superscript b Baseline x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript Subscript Superscript b minus a Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis Superscript b Baseline EndFraction 4th Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline left parenthesis x Subscript m Superscript b minus a Baseline plus sigma summation Underscript h equals 1 Overscript k Endscripts left parenthesis negative 1 right parenthesis Superscript h Baseline sigma summation Underscript 1 less than or equals i 1 less than midline horizontal ellipsis less than i Subscript h Baseline less than or equals k Endscripts x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript Subscript Superscript b minus a Baseline right parenthesis 5th Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline beta Subscript xi Sub Subscript b minus a Baseline left parenthesis k comma x Subscript m Baseline right parenthesis 6th Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Sub Subscript 1 ellipsis k Subscript Superscript b minus a Baseline product Underscript j equals 1 Overscript k Endscripts left parenthesis left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 right parenthesis period EndLayout

Subsequently, we calculate $E_2$ . From Lemmas 2.1, 2.2(i)–(iii), and 2.6, we can deduce that

$$ \begin{align*}E_2&=\sum_{t=1}^{s-k}(-1)^{t}\sum\limits_{k+1 \le j_1<\cdots <j_t \le s}\frac{[x_l,x_{m_{j_1\dots j_t}}]^b}{x_{m_{j_1\dots j_t}}^a} \\[2pt]&\quad+\sum_{h=1}^{k}\sum_{t=1}^{s-k}(-1)^{h+t}\sum\limits_{1 \le i_1<\cdots<i_h\le k \atop k+1 \le j_1<\cdots <j_t\le s}\frac{[x_l,x_{m_{i_1\dots i_hj_1\dots j_t}}]^b}{x_{m_{i_1\dots i_hj_1\dots j_t}}^a} \\[2pt]&=\sum_{t=1}^{s-k}(-1)^{t}\sum\limits_{k+1\le j_1<\cdots <j_t \le s}\frac{[x_l,x_m]^b}{x_{m_{j_1\dots j_t}}^a} \\[2pt]&\quad+\sum_{h=1}^{k}\sum_{t=1}^{s-k}(-1)^{h+t}\sum\limits_{1\le i_1<\cdots<i_h\le k \atop k+1\le j_1<\cdots <j_t\le s}\frac{[x_l,x_{m_{i_1\dots i_h}}]^b}{x_{m_{i_1\dots i_hj_1\dots j_t}}^a} \\[2pt]&=\sum_{t=1}^{s-k}(-1)^{t}\sum\limits_{k+1\le j_1<\cdots <j_t\le s}\frac{x_l^bx_m^b}{(x_l,x_m)^bx_{m_{j_1\dots j_t}}^a} \\[2pt]&\quad+\sum_{h=1}^{k}\sum_{t=1}^{s-k}(-1)^{h+t}\sum\limits_{1\le i_1<\cdots<i_h\le k \atop k+1\le j_1<\cdots <j_t\le s}\frac{x_l^bx_{m_{i_1\dots i_h}}^b}{(x_l,x_{m_{i_1\dots i_h}})^bx_{m_{i_1\dots i_hj_1\dots j_t}}^a} \\[2pt]&=\sum_{t=1}^{s-k}(-1)^{t}\sum\limits_{k+1\le j_1<\cdots <j_t\le s}\frac{x_l^bx_m^b}{(x_l,x_m)^bx_{m_{j_1\dots j_t}}^a} \\[2pt]&\quad+\sum_{h=1}^{k}\sum_{t=1}^{s-k}(-1)^{h+t}\sum\limits_{1\le i_1<\cdots<i_h\le k\atop k+1 \le j_1<\cdots <j_t\le s}\frac{x_l^bx_{m_{i_1\dots i_h}}^b}{(x_l,x_m)^bx_{m_{i_1\dots i_hj_1\dots j_t}}^a} \\&=\Big(\frac{x_l}{(x_l,x_m)}\Big)^b\Big(\sum_{t=1}^{s-k}(-1)^{t}\sum\limits_{k+1\le j_1<\cdots <j_t\le s}\frac{x_m^b}{\frac{x_m^a}{(P_{j_1}\dots P_{j_t})^a}} \\&\quad+\sum_{h=1}^{k}\sum_{t=1}^{s-k}(-1)^{h+t}\sum\limits_{1\le i_1<\cdots<i_h\le k \atop k+1\le j_1<\cdots <j_t \le s}\frac{\frac{x_m^b}{(P_{i_1}\dots P_{i_h})^b}}{\frac{x_m^a}{(P_{i_1}\dots P_{i_h}P_{j_1}\dots P_{j_t})^a}}\Big) \\&=\Big(\frac{x_l}{(x_l,x_m)}\Big)^b\Big(\sum_{t=1}^{s-k}(-1)^{t}\sum\limits_{k+1\le j_1<\cdots <j_t \le s}x_m^{b-a}(P_{j_1}\dots P_{j_t})^a \\&\quad+\sum_{h=1}^{k}\sum_{t=1}^{s-k}(-1)^{h+t}\sum\limits_{1\le i_1<\cdots<i_h\le k\atop k+1\le j_1<\cdots <j_t\le s}\frac{x_m^{b-a}}{(P_{i_1}\dots P_{i_h})^{b-a}}\cdot (P_{j_1}\dots P_{j_t})^a\Big) \\&=\Big(\frac{x_l}{(x_l,x_m)}\Big)^b\Big(\sum_{t=1}^{s-k}(-1)^{t}\sum\limits_{k+1\le j_1<\cdots <j_t\le s}(P_{j_1}\dots P_{j_t})^a\Big(x_m^{b-a} \\&\quad+\sum_{h=1}^{k}(-1)^h\sum\limits_{1 \le i_1<\cdots<i_h\le k}\frac{x_m^{b-a}}{(P_{i_1}\dots P_{i_h})^{b-a}}\Big)\Big) \end{align*} $$

StartLayout 1st Row 1st Column upper E 2 2nd Column equals sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript t Baseline sigma summation Underscript k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s Endscripts ContinuedFraction left bracket x Subscript l Baseline comma x Subscript m Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Baseline right bracket Superscript b Baseline Over x Subscript m Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Superscript a Baseline 2nd Row 1st Column Blank 2nd Column plus sigma summation Underscript h equals 1 Overscript k Endscripts sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript h plus t Baseline sigma summation Underscript StartLayout 1st Row 1 less than or equals i 1 less than midline horizontal ellipsis less than i Subscript h Baseline less than or equals k 2nd Row k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s EndLayout Endscripts ContinuedFraction left bracket x Subscript l Baseline comma x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Baseline right bracket Superscript b Baseline Over x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Superscript a Baseline 3rd Row 1st Column Blank 2nd Column equals sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript t Baseline sigma summation Underscript k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s Endscripts ContinuedFraction left bracket x Subscript l Baseline comma x Subscript m Baseline right bracket Superscript b Baseline Over x Subscript m Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Superscript a Baseline 4th Row 1st Column Blank 2nd Column plus sigma summation Underscript h equals 1 Overscript k Endscripts sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript h plus t Baseline sigma summation Underscript StartLayout 1st Row 1 less than or equals i 1 less than midline horizontal ellipsis less than i Subscript h Baseline less than or equals k 2nd Row k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s EndLayout Endscripts ContinuedFraction left bracket x Subscript l Baseline comma x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript Subscript Baseline right bracket Superscript b Baseline Over x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Superscript a Baseline 5th Row 1st Column Blank 2nd Column equals sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript t Baseline sigma summation Underscript k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s Endscripts ContinuedFraction x Subscript l Superscript b Baseline x Subscript m Superscript b Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis Superscript b Baseline x Subscript m Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Superscript a Baseline 6th Row 1st Column Blank 2nd Column plus sigma summation Underscript h equals 1 Overscript k Endscripts sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript h plus t Baseline sigma summation Underscript StartLayout 1st Row 1 less than or equals i 1 less than midline horizontal ellipsis less than i Subscript h Baseline less than or equals k 2nd Row k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s EndLayout Endscripts ContinuedFraction x Subscript l Superscript b Baseline x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript Subscript Superscript b Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript Subscript Baseline right parenthesis Superscript b Baseline x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Superscript a Baseline 7th Row 1st Column Blank 2nd Column equals sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript t Baseline sigma summation Underscript k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s Endscripts ContinuedFraction x Subscript l Superscript b Baseline x Subscript m Superscript b Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis Superscript b Baseline x Subscript m Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Superscript a Baseline 8th Row 1st Column Blank 2nd Column plus sigma summation Underscript h equals 1 Overscript k Endscripts sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript h plus t Baseline sigma summation Underscript StartLayout 1st Row 1 less than or equals i 1 less than midline horizontal ellipsis less than i Subscript h Baseline less than or equals k 2nd Row k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s EndLayout Endscripts ContinuedFraction x Subscript l Superscript b Baseline x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript Subscript Superscript b Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis Superscript b Baseline x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Superscript a Baseline 9th Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline left parenthesis sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript t Baseline sigma summation Underscript k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s Endscripts ContinuedFraction x Subscript m Superscript b Baseline Over StartFraction x Subscript m Superscript a Baseline Over left parenthesis upper P Subscript j 1 Baseline ellipsis upper P Subscript j Sub Subscript t Subscript Baseline right parenthesis Superscript a Baseline 10th Row 1st Column Blank 2nd Column plus sigma summation Underscript h equals 1 Overscript k Endscripts sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript h plus t Baseline sigma summation Underscript StartLayout 1st Row 1 less than or equals i 1 less than midline horizontal ellipsis less than i Subscript h Baseline less than or equals k 2nd Row k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s EndLayout Endscripts ContinuedFraction StartFraction x Subscript m Superscript b Baseline Over left parenthesis upper P Subscript i 1 Baseline ellipsis upper P Subscript i Sub Subscript h Subscript Baseline right parenthesis Superscript b Baseline Over StartFraction x Subscript m Superscript a Baseline Over left parenthesis upper P Subscript i 1 Baseline ellipsis upper P Subscript i Sub Subscript h Subscript Baseline upper P Subscript j 1 Baseline ellipsis upper P Subscript j Sub Subscript t Subscript Baseline right parenthesis Superscript a Baseline right parenthesis 11th Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline left parenthesis sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript t Baseline sigma summation Underscript k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s Endscripts x Subscript m Superscript b minus a Baseline left parenthesis upper P Subscript j 1 Baseline ellipsis upper P Subscript j Sub Subscript t Subscript Baseline right parenthesis Superscript a Baseline 12th Row 1st Column Blank 2nd Column plus sigma summation Underscript h equals 1 Overscript k Endscripts sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript h plus t Baseline sigma summation Underscript StartLayout 1st Row 1 less than or equals i 1 less than midline horizontal ellipsis less than i Subscript h Baseline less than or equals k 2nd Row k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s EndLayout Endscripts ContinuedFraction x Subscript m Superscript b minus a Baseline Over left parenthesis upper P Subscript i 1 Baseline ellipsis upper P Subscript i Sub Subscript h Subscript Baseline right parenthesis Superscript b minus a Baseline dot left parenthesis upper P Subscript j 1 Baseline ellipsis upper P Subscript j Sub Subscript t Subscript Baseline right parenthesis Superscript a Baseline right parenthesis 13th Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline left parenthesis sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript t Baseline sigma summation Underscript k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s Endscripts left parenthesis upper P Subscript j 1 Baseline ellipsis upper P Subscript j Sub Subscript t Subscript Baseline right parenthesis Superscript a Baseline left parenthesis x Subscript m Superscript b minus a Baseline 14th Row 1st Column Blank 2nd Column plus sigma summation Underscript h equals 1 Overscript k Endscripts left parenthesis negative 1 right parenthesis Superscript h Baseline sigma summation Underscript 1 less than or equals i 1 less than midline horizontal ellipsis less than i Subscript h Baseline less than or equals k Endscripts ContinuedFraction x Subscript m Superscript b minus a Baseline Over left parenthesis upper P Subscript i 1 Baseline ellipsis upper P Subscript i Sub Subscript h Subscript Baseline right parenthesis Superscript b minus a Baseline right parenthesis right parenthesis EndLayout

$$ \begin{align*}&=\Big(\frac{x_l}{(x_l,x_m)}\Big)^b\Big(\sum_{t=1}^{s-k}(-1)^{t}\sum\limits_{k+1 \le j_1<\cdots <j_t \le s}(P_{j_1}\dots P_{j_t})^a\Big(x_m^{b-a} \\&\quad+\sum_{h=1}^{k}(-1)^h\sum\limits_{1\le i_1<\cdots<i_h\le k}x_{m_{i_1\dots i_h}}^{b-a}\Big)\Big) \\&=\Big(\frac{x_l}{(x_l,x_m)}\Big)^b\Big(\sum_{t=1}^{s-k}(-1)^{t}\sum\limits_{k+1\le j_1<\cdots <j_t \le s}(P_{j_1}\dots P_{j_t})^a\beta_{\xi_{b-a}}(k,x_m)\Big) \\&=\Big(\frac{x_l}{(x_l,x_m)}\Big)^b\Big(\sum_{t=1}^{s-k}(-1)^{t}\sum\limits_{k+1\le j_1<\cdots <j_t \le s}(P_{j_1}\dots P_{j_t})^ax_{m_{1\dots k}}^{b-a} \\&\quad\times \prod_{j=1}^k\Big(\Big(\frac{x_m}{x_{m_j}}\Big)^{b-a}-1\Big)\Big) \\&=\Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_{m_{1\dots k}}^{b-a}\prod_{j=1}^k\Big(\Big(\frac{x_m}{x_{m_j}}\Big)^{b-a}-1\Big)\sum_{t=1}^{s-k}(-1)^{t}\sum\limits_{k+1\le j_1<\cdots <j_t \le s}(P_{j_1}\dots P_{j_t})^a \\&=\Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_{m_{1\dots k}}^{b-a}\prod_{j=1}^k\Big(\Big(\frac{x_m}{x_{m_j}}\Big)^{b-a}-1\Big)\sum_{t=1}^{s-k}(-1)^{t}\sum\limits_{k+1\le j_1<\cdots <j_t \le s}\frac{x_m^a}{\frac{x_m^a}{(P_{j_1}\dots P_{j_t})^a}} \\&=\Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_{m_{1\dots k}}^{b-a}\prod_{j=1}^k\Big(\Big(\frac{x_m}{x_{m_j}}\Big)^{b-a}-1\Big)\sum_{t=1}^{s-k}(-1)^{t}\sum\limits_{k+1 \le j_1<\cdots <j_t \le s}\frac{x_m^a}{x_{m_{j_1\dots j_t}}^a} \\&=\Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_{m_{1\dots k}}^{b-a}\prod_{j=1}^k\Big(\Big(\frac{x_m}{x_{m_j}}\Big)^{b-a}-1\Big) \\&\quad\times\Big(x_m^a\Big(x_m^{-a}+\sum_{t=1}^{s-k}(-1)^{t}\sum^{s-k}\limits_{k+1 \le j_1<\cdots <j_t \le s}x_{m_{j_1\dots j_t}}^{-a}\Big)-1\Big) \\&=\Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_{m_{1\dots k}}^{b-a}\prod_{j=1}^k\Big(\Big(\frac{x_m}{x_{m_j}}\Big)^{b-a}-1\Big)(x_m^a\beta_{\xi_{-a}}(s-k,x_m)-1) \\&=\Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_{m_{1\dots k}}^{b-a}\prod_{j=1}^k\Big(\Big(\frac{x_m}{x_{m_j}}\Big)^{b-a}-1\Big)\Big(x_m^a\Big(x_{m_{(k+1)\dots s}}^{-a}\prod_{j=k+1}^s\Big(\Big(\frac{x_m}{x_{m_j}}\Big)^{-a}-1\Big)-1\Big).\end{align*} $$

StartLayout 1st Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline left parenthesis sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript t Baseline sigma summation Underscript k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s Endscripts left parenthesis upper P Subscript j 1 Baseline ellipsis upper P Subscript j Sub Subscript t Subscript Baseline right parenthesis Superscript a Baseline left parenthesis x Subscript m Superscript b minus a Baseline 2nd Row 1st Column Blank 2nd Column plus sigma summation Underscript h equals 1 Overscript k Endscripts left parenthesis negative 1 right parenthesis Superscript h Baseline sigma summation Underscript 1 less than or equals i 1 less than midline horizontal ellipsis less than i Subscript h Baseline less than or equals k Endscripts x Subscript m Sub Subscript i 1 ellipsis i Sub Sub Subscript h Sub Subscript Subscript Superscript b minus a Baseline right parenthesis right parenthesis 3rd Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline left parenthesis sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript t Baseline sigma summation Underscript k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s Endscripts left parenthesis upper P Subscript j 1 Baseline ellipsis upper P Subscript j Sub Subscript t Subscript Baseline right parenthesis Superscript a Baseline beta Subscript xi Sub Subscript b minus a Subscript Baseline left parenthesis k comma x Subscript m Baseline right parenthesis right parenthesis 4th Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline left parenthesis sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript t Baseline sigma summation Underscript k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s Endscripts left parenthesis upper P Subscript j 1 Baseline ellipsis upper P Subscript j Sub Subscript t Subscript Baseline right parenthesis Superscript a Baseline x Subscript m Sub Subscript 1 ellipsis k Subscript Superscript b minus a Baseline 5th Row 1st Column Blank 2nd Column times product Underscript j equals 1 Overscript k Endscripts left parenthesis left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 right parenthesis right parenthesis 6th Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Sub Subscript 1 ellipsis k Superscript b minus a Baseline product Underscript j equals 1 Overscript k Endscripts left parenthesis left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 right parenthesis sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript t Baseline sigma summation Underscript k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s Endscripts left parenthesis upper P Subscript j 1 Baseline ellipsis upper P Subscript j Sub Subscript t Subscript Baseline right parenthesis Superscript a Baseline 7th Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Sub Subscript 1 ellipsis k Superscript b minus a Baseline product Underscript j equals 1 Overscript k Endscripts left parenthesis left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 right parenthesis sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript t Baseline sigma summation Underscript k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s Endscripts ContinuedFraction x Subscript m Superscript a Baseline Over StartFraction x Subscript m Superscript a Baseline Over left parenthesis upper P Subscript j 1 Baseline ellipsis upper P Subscript j Sub Subscript t Subscript Baseline right parenthesis Superscript a Baseline 8th Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Sub Subscript 1 ellipsis k Superscript b minus a Baseline product Underscript j equals 1 Overscript k Endscripts left parenthesis left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 right parenthesis sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript t Baseline sigma summation Underscript k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s Endscripts ContinuedFraction x Subscript m Superscript a Baseline Over x Subscript m Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Superscript a Baseline 9th Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Sub Subscript 1 ellipsis k Superscript b minus a Baseline product Underscript j equals 1 Overscript k Endscripts left parenthesis left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 right parenthesis 10th Row 1st Column Blank 2nd Column times left parenthesis x Subscript m Superscript a Baseline left parenthesis x Subscript m Superscript negative a Baseline plus sigma summation Underscript t equals 1 Overscript s minus k Endscripts left parenthesis negative 1 right parenthesis Superscript t Baseline sigma summation Underscript k plus 1 less than or equals j 1 less than midline horizontal ellipsis less than j Subscript t Baseline less than or equals s Overscript s minus k Endscripts x Subscript m Sub Subscript j 1 ellipsis j Sub Sub Subscript t Sub Subscript Subscript Superscript negative a Baseline right parenthesis minus 1 right parenthesis 11th Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Sub Subscript 1 ellipsis k Superscript b minus a Baseline product Underscript j equals 1 Overscript k Endscripts left parenthesis left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 right parenthesis left parenthesis x Subscript m Superscript a Baseline beta Subscript xi Sub Subscript negative a Subscript Baseline left parenthesis s minus k comma x Subscript m Baseline right parenthesis minus 1 right parenthesis 12th Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Sub Subscript 1 ellipsis k Subscript Superscript b minus a Baseline product Underscript j equals 1 Overscript k Endscripts left parenthesis left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 right parenthesis left parenthesis x Subscript m Superscript a Baseline left parenthesis x Subscript m Sub Subscript left parenthesis k plus 1 right parenthesis ellipsis s Subscript Superscript negative a Baseline product Underscript j equals k plus 1 Overscript s Endscripts left parenthesis left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript negative a Baseline minus 1 right parenthesis minus 1 right parenthesis period EndLayout

Thus,

$$ \begin{align*}E_1+E_2=\Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_{m_{1\dots k}}^{b-a}x_m^ax_{m_{(k+1)\dots s}}^{-a} \prod\limits_{j=1}^k\Big(\Big(\frac{x_m}{x_{m_j}}\Big)^{b-a}-1\Big)\prod\limits_{j=k+1}^s\Big(\Big(\frac{x_m}{x_{m_j}}\Big)^{-a}-1\Big), \end{align*} $$

and so by Lemmas 2.1 and 2.7, we have

$$ \begin{align*} H(l,m)&=\frac{E_1+E_2}{\alpha_{\frac{1}{\xi_a}}(x_m)}\\ &=\Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_m^ax_{m_{1\dots k}}^{b-a}x_{m_{(k+1)\dots s}}^{-a}x_{m_{1\dots s}}^a \frac{\prod_{j=1}^k\big(\big(\frac{x_m}{x_{m_j}}\big)^{b-a}-1\big)}{\prod_{j=1}^k\big(\big(\frac{x_m}{x_{m_j}}\big)^{-a}-1\big)}\\ &=\Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_m^ax_{m_{1\dots k}}^{b-a}x_{m_{(k+1)\dots s}}^{-a}x_{m_{1\dots s}}^a\\ &\quad\times(-1)^k\Big(\prod_{j=1}^k\frac{x_m}{x_{m_j}}\Big)^a\frac{\prod_{j=1}^k\big(\big(\frac{x_m}{x_{m_j}}\big)^{b-a}-1\big)} {\prod_{j=1}^k\big(\big(\frac{x_m}{x_{m_j}}\big)^a-1\big)}\\ &=\Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_m^ax_{m_{1\dots k}}^{b-a}x_{m_{(k+1)\dots s}}^{-a}x_{m_{1\dots s}}^a\\ &\quad\times(-1)^k(P_1\dots P_k)^a\prod_{j=1}^k\frac{\big(\frac{x_m}{x_{m_j}}\big)^{b-a}-1}{\big(\frac{x_m}{x_{m_j}}\big)^a-1}\\ &=\Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_m^ax_{m_{1\dots k}}^{b-a}\frac{(P_{k+1}\dots P_s)^a}{x_m^a}x_{m_{1\dots s}}^a\\ &\quad\times(-1)^k(P_1\dots P_k)^a\prod_{j=1}^k\frac{\big(\frac{x_m}{x_{m_j}}\big)^{b-a}-1}{\big(\frac{x_m}{x_{m_j}}\big)^a-1}\\ &=\Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_m^ax_{m_{1\dots k}}^{b-a}(-1)^k \prod_{j=1}^k\frac{\big(\frac{x_m}{x_{m_j}}\big)^{b-a}-1}{\big(\frac{x_m}{x_{m_j}}\big)^a-1}\\ &= (-1)^k \Big(\frac{x_l}{(x_l,x_m)}\Big)^bx_m^ax_{m_{1\dots k}}^{b-a} \prod_{j=1}^k\frac{\big(\frac{x_m}{x_{m_j}}\big)^{b-a}-1}{\big(\frac{x_m}{x_{m_j}}\big)^a-1} \end{align*} $$

StartLayout 1st Row 1st Column upper H left parenthesis l comma m right parenthesis 2nd Column equals StartFraction upper E 1 plus upper E 2 Over alpha Subscript StartFraction 1 Over xi Sub Subscript a Subscript EndFraction Baseline left parenthesis x Subscript m Baseline right parenthesis EndFraction 2nd Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Superscript a Baseline x Subscript m Sub Subscript 1 ellipsis k Superscript b minus a Baseline x Subscript m Sub Subscript left parenthesis k plus 1 right parenthesis ellipsis s Superscript negative a Baseline x Subscript m Sub Subscript 1 ellipsis s Superscript a Baseline StartFraction product Underscript j equals 1 Overscript k Endscripts left parenthesis left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 right parenthesis Over product Underscript j equals 1 Overscript k Endscripts left parenthesis left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript negative a Baseline minus 1 right parenthesis EndFraction 3rd Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Superscript a Baseline x Subscript m Sub Subscript 1 ellipsis k Superscript b minus a Baseline x Subscript m Sub Subscript left parenthesis k plus 1 right parenthesis ellipsis s Superscript negative a Baseline x Subscript m Sub Subscript 1 ellipsis s Superscript a Baseline 4th Row 1st Column Blank 2nd Column times left parenthesis negative 1 right parenthesis Superscript k Baseline left parenthesis product Underscript j equals 1 Overscript k Endscripts StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript a Baseline StartFraction product Underscript j equals 1 Overscript k Endscripts left parenthesis left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 right parenthesis Over product Underscript j equals 1 Overscript k Endscripts left parenthesis left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript a Baseline minus 1 right parenthesis EndFraction 5th Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Superscript a Baseline x Subscript m Sub Subscript 1 ellipsis k Superscript b minus a Baseline x Subscript m Sub Subscript left parenthesis k plus 1 right parenthesis ellipsis s Superscript negative a Baseline x Subscript m Sub Subscript 1 ellipsis s Superscript a Baseline 6th Row 1st Column Blank 2nd Column times left parenthesis negative 1 right parenthesis Superscript k Baseline left parenthesis upper P 1 ellipsis upper P Subscript k Baseline right parenthesis Superscript a Baseline product Underscript j equals 1 Overscript k Endscripts StartFraction left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 Over left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript a Baseline minus 1 EndFraction 7th Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Superscript a Baseline x Subscript m Sub Subscript 1 ellipsis k Superscript b minus a Baseline StartFraction left parenthesis upper P Subscript k plus 1 Baseline ellipsis upper P Subscript s Baseline right parenthesis Superscript a Baseline Over x Subscript m Superscript a Baseline EndFraction x Subscript m Sub Subscript 1 ellipsis s Superscript a Baseline 8th Row 1st Column Blank 2nd Column times left parenthesis negative 1 right parenthesis Superscript k Baseline left parenthesis upper P 1 ellipsis upper P Subscript k Baseline right parenthesis Superscript a Baseline product Underscript j equals 1 Overscript k Endscripts StartFraction left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 Over left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript a Baseline minus 1 EndFraction 9th Row 1st Column Blank 2nd Column equals left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Superscript a Baseline x Subscript m Sub Subscript 1 ellipsis k Superscript b minus a Baseline left parenthesis negative 1 right parenthesis Superscript k Baseline product Underscript j equals 1 Overscript k Endscripts StartFraction left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 Over left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript a Baseline minus 1 EndFraction 10th Row 1st Column Blank 2nd Column equals left parenthesis negative 1 right parenthesis Superscript k Baseline left parenthesis StartFraction x Subscript l Baseline Over left parenthesis x Subscript l Baseline comma x Subscript m Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript m Superscript a Baseline x Subscript m Sub Subscript 1 ellipsis k Superscript b minus a Baseline product Underscript j equals 1 Overscript k Endscripts StartFraction left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 Over left parenthesis StartFraction x Subscript m Baseline Over x Subscript m Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript a Baseline minus 1 EndFraction EndLayout

as asserted.

This finishes the proof of Lemma 3.1.

Finally, we conclude this article by giving the proof of Theorem 1.2.

Proof of Theorem 1.2

Let S be a gcd-closed set which satisfies the condition $\mathcal {G}$ and let a and b be positive integers with $a\mid b$ . For any integers i and j with $1 \le i,j \le |S|$ , by Lemma 2.8, we have

$$ \begin{align*} {\left([S^b][S^a]^{-1}\right)}_{ij} =&\sum_{r=1}^{|S|}[x_i, x_r]^b\dfrac{1}{x_r^ax_j^a}\sum_{x_r|x_t\atop x_j|x_t} \dfrac{c_{rt}c_{jt}}{\alpha_{\frac{1}{\xi_a}}(x_t)}\\ =&\sum\limits_{x_j|x_t}c_{jt}\frac{1}{x_j^a\alpha_{\frac{1}{\xi_a}}(x_t)}\sum\limits_{x_r|x_t} \frac{c_{rt}[x_i, x_r]^b}{x_r^a}\\ =&\sum\limits_{x_j|x_t}c_{jt}\frac{H(i,t)}{x_j^a} \end{align*} $$

StartLayout 1st Row 1st Column left parenthesis left bracket upper S Superscript b Baseline right bracket left bracket upper S Superscript a Baseline right bracket Superscript negative 1 Baseline right parenthesis Subscript i j Baseline equals 2nd Column sigma summation Underscript r equals 1 Overscript StartAbsoluteValue upper S EndAbsoluteValue Endscripts left bracket x Subscript i Baseline comma x Subscript r Baseline right bracket Superscript b Baseline StartFraction 1 Over x Subscript r Superscript a Baseline x Subscript j Superscript a Baseline EndFraction sigma summation Underscript StartLayout 1st Row x Subscript r Baseline vertical bar x Subscript t Baseline 2nd Row x Subscript j Baseline vertical bar x Subscript t Baseline EndLayout Endscripts StartFraction c Subscript r t Baseline c Subscript j t Baseline Over alpha Subscript StartFraction 1 Over xi Sub Subscript a Subscript EndFraction Baseline left parenthesis x Subscript t Baseline right parenthesis EndFraction 2nd Row 1st Column equals 2nd Column sigma summation Underscript x Subscript j Baseline vertical bar x Subscript t Baseline Endscripts c Subscript j t Baseline StartFraction 1 Over x Subscript j Superscript a Baseline alpha Subscript StartFraction 1 Over xi Sub Subscript a Subscript EndFraction Baseline left parenthesis x Subscript t Baseline right parenthesis EndFraction sigma summation Underscript x Subscript r Baseline vertical bar x Subscript t Baseline Endscripts StartFraction c Subscript r t Baseline left bracket x Subscript i Baseline comma x Subscript r Baseline right bracket Superscript b Baseline Over x Subscript r Superscript a Baseline EndFraction 3rd Row 1st Column equals 2nd Column sigma summation Underscript x Subscript j Baseline vertical bar x Subscript t Baseline Endscripts c Subscript j t Baseline StartFraction upper H left parenthesis i comma t right parenthesis Over x Subscript j Superscript a Baseline EndFraction EndLayout

with $H(i,t)$ being defined as in (3.1).

In what follows, we prove that

$$ \begin{align*}{\left( [S^b][S^a]^{-1}\right)}_{ij}\in\mathbb Z\end{align*} $$

for all integers i and j with $1\le i,j \le |S|$ . By the definition of $c_{jt} $ as in (2.2), we know that $ c_{jt} \in \mathbb {Z}$ . So we need only to show that for any positive integer t with $x_j\mid x_t$ , we have

$$ \begin{align*}\frac{H(i,t)}{x_j^a}\in \mathbb{Z}.\end{align*} $$ StartFraction upper H left parenthesis i comma t right parenthesis Over x Subscript j Superscript a Baseline EndFraction element of double struck upper Z period

This will be shown in what follows.

For any given integers i and t with $1 \le i,t\le |S|$ and $x_j|x_t$ , we consider the following two cases.

Case 1. $t=1$ . Then $x_t=x_1=\gcd (S)$ . At this moment, we must have $x_j=x_1$ . By Lemmas 2.4 and 2.9, we obtain that

$$ \begin{align*} \frac{H(i,t)}{x_j^a}=\frac{H(i,1)}{x_1^a}=\frac{1}{x_1^a\alpha_{\frac{1}{\xi_a}}(x_1)}\sum\limits_{x_r|x_1}\frac{c_{r1} [x_i, x_r]^b}{x_r^a}=\frac{[x_i,x_1]^b}{x_1^a}=\frac{x_i^b}{x_1^a}=x_i^{b-a}\Big(\frac{x_i}{x_1}\Big)^a\in \mathbb Z \end{align*} $$

as one expects.

Case 2. $t\ge 2$ . Since S is gcd closed, one has $x_1\mid x_t$ and so $|G_S(x_t)|\ge 1$ . Write $G_S(x_t):=\{x_{t_1},\ldots , x_{t_s}\}$ with $s\ge 1$ . Since $a\mid b$ and S satisfies the condition $\mathcal {G}$ as well as $x_j\mid x_t$ , by Lemma 3.1, we have

$$ \begin{align*} \frac{H(i,t)}{x_j^a}=\left\{ \begin{aligned} &(-1)^s\Big(\frac{x_i}{(x_i,x_t)}\Big)^bx_{t_{1\dots s}}^{b-a}\Big(\frac{x_t}{x_j}\Big)^a \prod_{j=1}^s\frac{\big(\frac{x_t}{x_{t_j}}\big)^{b-a}-1}{\big(\frac{x_t}{x_{t_j}}\big)^a-1}\in\mathbb Z, \ \ \text{if}\ (x_i,x_t)\mid x_{t_{1\dots s}},\\ &x_i^{b-a}\Big(\frac{x_i}{x_j}\Big)^a\in\mathbb Z, \ \ \text{if}\ x_t\mid x_i,\\ &(-1)^k\Big(\frac{x_i}{(x_i,x_t)}\Big)^b\Big(\frac{x_t}{x_j}\Big)^ax_{t_{1\dots k}}^{b-a} \prod_{j=1}^k\frac{\big(\frac{x_t}{x_{t_j}}\big)^{b-a}-1}{\big(\frac{x_t}{x_{t_j}}\big)^a-1}\in\mathbb Z, \ \text{if}\ (x_i,x_t)\mid x_{t_{1\dots k}}\\ &\text{for some}\ 1\le k\le s-1\ \text{and}\ (x_i,x_t) \nmid x_{t_j}\ \text{for all}\ k+1 \le j \le s. \end{aligned} \right. \end{align*} $$

StartFraction upper H left parenthesis i comma t right parenthesis Over x Subscript j Superscript a Baseline EndFraction equals StartLayout Enlarged left brace 1st Row 1st Column Blank 2nd Column left parenthesis negative 1 right parenthesis Superscript s Baseline left parenthesis StartFraction x Subscript i Baseline Over left parenthesis x Subscript i Baseline comma x Subscript t Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline x Subscript t Sub Subscript 1 ellipsis s Subscript Superscript b minus a Baseline left parenthesis StartFraction x Subscript t Baseline Over x Subscript j Baseline EndFraction right parenthesis Superscript a Baseline product Underscript j equals 1 Overscript s Endscripts StartFraction left parenthesis StartFraction x Subscript t Baseline Over x Subscript t Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 Over left parenthesis StartFraction x Subscript t Baseline Over x Subscript t Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript a Baseline minus 1 EndFraction element of double struck upper Z comma if left parenthesis x Subscript i Baseline comma x Subscript t Baseline right parenthesis vertical bar x Subscript t Sub Subscript 1 ellipsis s Subscript Baseline comma 2nd Row 1st Column Blank 2nd Column x Subscript i Superscript b minus a Baseline left parenthesis StartFraction x Subscript i Baseline Over x Subscript j Baseline EndFraction right parenthesis Superscript a Baseline element of double struck upper Z comma if x Subscript t Baseline vertical bar x Subscript i Baseline comma 3rd Row 1st Column Blank 2nd Column left parenthesis negative 1 right parenthesis Superscript k Baseline left parenthesis StartFraction x Subscript i Baseline Over left parenthesis x Subscript i Baseline comma x Subscript t Baseline right parenthesis EndFraction right parenthesis Superscript b Baseline left parenthesis StartFraction x Subscript t Baseline Over x Subscript j Baseline EndFraction right parenthesis Superscript a Baseline x Subscript t Sub Subscript 1 ellipsis k Subscript Superscript b minus a Baseline product Underscript j equals 1 Overscript k Endscripts StartFraction left parenthesis StartFraction x Subscript t Baseline Over x Subscript t Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript b minus a Baseline minus 1 Over left parenthesis StartFraction x Subscript t Baseline Over x Subscript t Sub Subscript j Subscript Baseline EndFraction right parenthesis Superscript a Baseline minus 1 EndFraction element of double struck upper Z comma if left parenthesis x Subscript i Baseline comma x Subscript t Baseline right parenthesis vertical bar x Subscript t Sub Subscript 1 ellipsis k Subscript Baseline 4th Row 1st Column Blank 2nd Column for some 1 less than or equals k less than or equals s minus 1 and left parenthesis x Subscript i Baseline comma x Subscript t Baseline right parenthesis does not divide x Subscript t Sub Subscript j Subscript Baseline for all k plus 1 less than or equals j less than or equals s period EndLayout

It then follows that

$$ \begin{align*}\frac{H(i,t)}{x_j^a}\in\mathbb Z\end{align*} $$ StartFraction upper H left parenthesis i comma t right parenthesis Over x Subscript j Superscript a Baseline EndFraction element of double struck upper Z

as required.

This concludes the proof of Theorem 1.2.

Acknowledgements

The author is grateful to the anonymous referee for insightful and invaluable suggestions and comments that improved greatly the presentation of the article.

Footnotes

The research was supported in part by the Startup Research Fund of Hubei Minzu University for Doctoral Scholars (Grant No. BS25008)

References

Apostol, T. M., Arithmetic properties of generalized Ramanujan sums . Pacific J. Math. 41(1972), 281–293.Google Scholar

Beslin, S. and Ligh, S., Another generalization of Smith’s determinant . Bull. Aust. Math. Soc. 40(1989), 413–415.Google Scholar

Bhatia, R. and Dias da Silva, J. A., Infinite divisibility of GCD matrices . Amer. Math. Monthly 115(2008), 551–553.Google Scholar

Bourque, K. and Ligh, S., On GCD and LCM matrices . Linear Algebra Appl. 174(1992), 65–74.Google Scholar

Bourque, K. and Ligh, S., Matrices associated with classes of arithmetical functions . J. Number Theory 45(1993), 367–376.Google Scholar

Bourque, K. and Ligh, S., Matrices associated with arithmetical functions . Linear Multilinear Algebra 34(1993), 261–267.Google Scholar

Bourque, K. and Ligh, S., Matrices associated with multiplicative functions . Linear Algebra Appl. 216(1995), 267–275.Google Scholar

Feng, W. D., Hong, S. F., and Zhao, J. R., Divisibility properties of power LCM matrices by power GCD matrices on gcd-closed sets . Discrete Math. 309(2009), 2627–2639.Google Scholar

Hilberdink, T., Determinants of multiplicative Toeplitz matrices . Acta Arith. 125(2006), 265–284.Google Scholar

Hong, S. A., Hu, S. N., and Lin, Z. B., On a certain arithmetical determinant . Acta Math. Hungar. 150(2016), 372–382.Google Scholar

Hong, S. A. and Lin, Z. B., New results on the value of a certain arithmetical determinant . Publ. Math. Debrecen 93(2018), 171–187.Google Scholar

Hong, S. F., On the Bourque-Ligh conjecture of least common multiple matrices . J. Algebra 218(1999), 216–228.Google Scholar

Hong, S. F., Gcd-closed sets and determinants of matrices associated with arithmetical functions . Acta Arith. 101(2002), 321–332.Google Scholar

Hong, S. F., On the factorization of LCM matrices on gcd-closed sets . Linear Algebra Appl. 345(2002), 225–233.Google Scholar

Hong, S. F., Factorization of matrices associated with classes of arithmetical functions . Colloq. Math. 98(2003), 113–123.Google Scholar

Hong, S. F., Nonsingularity of matrices associated with classes of arithmetical functions . J. Algebra 281(2004), 1–14.Google Scholar

Hong, S. F., Divisibility properties of power GCD matrices and power LCM matrices . Linear Algebra Appl. 428(2008), 1001–1008.Google Scholar

Hong, S. F., Divisibility among power GCD matrices and power LCM matrices . Bull. Aust. Math. Soc. 113(2026), 231–243.Google Scholar

Hong, S. F. and Enoch Lee, K. S., Asymptotic behavior of eigenvalues of reciprocal power LCM matrices . Glasg. Math. J. 50(2008), 163–174.Google Scholar

Hong, S. F. and Loewy, R., Asymptotic behavior of eigenvalues of greatest common divisor matrices . Glasg. Math. J. 46(2004), 551–569.Google Scholar

Hong, S. F. and Loewy, R., Asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions $\left(\mathit{\operatorname{mod}}\;r\right)$

. Int. J. Number Theory 7(2011), 1681–1704.Google Scholar

Hong, S. F., Zhao, J., and Yin, Y. Z., Divisibility properties of Smith matrices . Acta Arith. 132(2008), 161–175.Google Scholar

Li, M. and Tan, Q. R., Divisibility of matrices associated with multiplicative functions . Discret. Math. 311(2011), 2276–2282.Google Scholar

Lin, Z. B. and Hong, S. A., More on a certain arithmetical determinant . Bull. Aust. Math. Soc. 97(2018), 15–25.Google Scholar

McCarthy, P. J., A generalization of Smith’s determinant . Canad. Math. Bull. 29(1986), 109–113.Google Scholar

Smith, H. J. S., On the value of a certain arithmetical determinant . Proc. Lond. Math. Soc. 7(1875–1876), 208–212.Google Scholar

Tan, Q. R. and Li, M., Divisibility among power GCD matrices and among power LCM matrices on finitely many coprime divisor chains . Linear Algebra Appl. 438(2013), 1454–1466.Google Scholar

Wan, J. X. and Zhu, G. Y., Divisibility among power GCD and power LCM matrices on certain gcd-closed sets . Linear Multilinear Algebra 74(2026), 144–165.Google Scholar

Yamasaki, Y., Arithmetical properties of multiple Ramanujan sums . Ramanujan J. 21(2010), 241–261.Google Scholar

Zhu, G. Y., On the divisibility among power GCD and power LCM matrices on gcd-closed sets . Int. J. Number Theory 18(2022), 1397–1408.Google Scholar

Zhu, G. Y., On a certain determinant for a U.F.D . Colloq. Math. 171(2023), 49–59.Google Scholar

Zhu, G. Y., Gcd-closed sets and divisibility among power GCD matrices and power LCM matrices . Period. Math. Hungar. in press.Google Scholar

Zhu, G. Y. and Li, M., On the divisibility among power LCM matrices on gcd-closed sets . Bull. Aust. Math. Soc. 107(2023), 31–39.Google Scholar

Zhu, G. Y., Luo, Y. Y., and Wan, J. X., Factorization of power GCD matrices and power LCM matrices on certain gcd-closed sets . Int. J. Number Theory. https://doi.org/10.1142/S1793042126501204 Google Scholar

Zhu, G. Y. and Yu, J. M., Proof of a conjecture of Hong on nonsingularity of power LCM matrices on gcd-closed sets . Proc. Amer. Math. Soc. 154(2026), 1911–1925.Google Scholar

Article contents

GCD-closed sets and divisibility among power LCM matrices

Abstract

Keywords

MSC classification

Information

1 Introduction

Conjecture 1.1 [Reference Hong18, Conjecture 3.4]

2 Preliminary lemmas

Lemma 2.1 [Reference Zhu and Yu35]

Lemma 2.2 [Reference Zhu32]

Lemma 2.3 [Reference Bourque and Ligh6, Theorem 3]

Lemma 2.4 [Reference Hong16]

Lemma 2.5 [Reference Bourque and Ligh6]

Lemma 2.6 [Reference Zhu and Yu35]

Lemma 2.9 [Reference Zhu32]

3 A key lemma and proof of Theorem 1.2

Proof of Theorem 1.2

Acknowledgements

Footnotes

References

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests