Determinants of matrices associated with arithmetic functions on finitely many quasi-coprime divisor chains

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Abstract

Let S={x1,,xn} be a set of n distinct positive integers and f be an arithmetic function. We use f(S)=f(xi,xj) (resp. f[S]=f[xi,xj]) to denote the n×n matrix having f evaluated at the greatest common divisor (resp. the least common multiple) of xi and xj as its i,j-entry. The set S is called a divisor chain if there is a permutation σ of {1,,n} such that xσ(1)||xσ(n). If S can be partitioned as S=i=1kSi with all Si(1ik) being divisor chains and (max(Si), max(Sj)) = gcd(S) for 1ijk, then we say that S consists of finitely many quasi-coprime divisor chains. In this paper, we introduce a new method to give the formulas for the determinants of the matrices (f(S)) and (f[S]) on finitely many quasi-coprime divisor chains S. We show also that det(f(S))|det(f[S]) holds under some natural conditions. These extend the results obtained by Tan and Lin (2010) and Tan et al. (2013), respectively.

Section snippets

Introduction and statements of main results

Let S={x1,,xn} be a set of n distinct positive integers and f be an arithmetic function. We use f(S)=f(xi,xj) (resp. f[S]=f[xi,xj]) to denote the n×n matrix having f evaluated at the greatest common divisor (resp. the least common multiple) of xi and xj as its i,j-entry. If f(x)=x, then f(S)=S (resp. f[S]=[S]) is simply called GCD matrix (resp. LCM matrix). In 1875, Smith [14] first published his famous result by showing that det(S)=i=1nφ(xi) and det[S]=i=1n[φ(xi)p|xi(-p)] when S is factor

Proofs of Theorems 1.1 and 1.2 and Corollary 1.2

In this section, our main goal is to prove Theorem 1.1, Theorem 1.2. First we give three known lemmas.

Lemma 2.1

[13]

Let f be a multiplicative function and m and n be any positive integers. Then f(m)f(n)=f((m,n))f([m,n]).

Recall from [9] that for x,yS, and x<y, if x|y and the conditions x|d|y and dS imply that d{x,y}, then we say that x is a greatest-type divisor of y in S. We use GS(x) to denote the set of all greatest-type divisors of x in S.

Lemma 2.2

[4], [9]

Let S be gcd closed and f be an arithmetic function. Then detf(S)

Examples

In this section, we give some examples to demonstrate our main results.

Example 3.1

Let λ be the Liouville function which is defined byλ(n)=1,ifn=1,(-1)α1++αr,ifn=p1α1p1αr,p1,,prare distinct positive primes.Letting S={2,4,8,6,18,} gives us(λ(S))=-1-1-1-1-1-111-1-1-11-1-1-1-1-1-111-1-1-11-1,(λ[S])=-11-11-111-1-11-1-1-11-11-111-1-11-1-1-1.Clearly, S consists of two quasi-coprime divisor chains such that gcd(S)=2S. Then by Theorem 1.1, we can easily get thatdet(λ(S))=λ(2)(λ(4)-λ(2))(λ(8)-λ(4))(λ(6)-λ(2))(λ(

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Zhao is the corresponding author and Hong was supported partially by National Science Foundation of China Grant #11371260.

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