Determinants of matrices associated with arithmetic functions on finitely many quasi-coprime divisor chains☆
Section snippets
Introduction and statements of main results
Let be a set of n distinct positive integers and f be an arithmetic function. We use (resp. ) to denote the matrix having f evaluated at the greatest common divisor (resp. the least common multiple) of and as its -entry. If , then (resp. ) is simply called GCD matrix (resp. LCM matrix). In 1875, Smith [14] first published his famous result by showing that = and = 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 Let f be a multiplicative function and m and n be any positive integers. Then .[13]
Recall from [9] that for , and , if and the conditions and imply that , then we say that x is a greatest-type divisor of y in S. We use to denote the set of all greatest-type divisors of x in S. Lemma 2.2 Let S be gcd closed and f be an arithmetic function. Then [4], [9]
Examples
In this section, we give some examples to demonstrate our main results. Example 3.1 Let be the Liouville function which is defined byLetting gives usClearly, S consists of two quasi-coprime divisor chains such that gcd. Then by Theorem 1.1, we can easily get that
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