Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions

Abstract Let f be an arithmetic function and S = {x1, …, xn} be a set of n distinct positive integers. By (f(xi, xj)) (resp. (f[xi, xj])) we denote the n × n matrix having f evaluated at the greatest common divisor (xi, xj) (resp. the least common multiple [xi, xj]) of x, and xj as its (i, j)-entry, respectively. The set S is said to be gcd closed if (xi, xj) ∈ S for 1 ≤ i, j ≤ n. In this paper, we give formulas for the determinants of the matrices (f(xi, xj)) and (f[xi, xj]) if S consists of multiple coprime gcd-closed sets (i.e., S equals the union of S1, …, Sk with k ≥ 1 being an integer and S1, …, Sk being gcd-closed sets such that (lcm(Si), lcm(Sj)) = 1 for all 1 ≤ i ≠ j ≤ k). This extends the Bourque-Ligh, Hong’s and the Hong-Loewy formulas obtained in 1993, 2002 and 2011, respectively. It also generalizes the famous Smith’s determinant.


Introduction and statements of main results
Let n be a positive integer and f be an arithmetic function. Let S D fx 1 ; :::; x n g be a set of n distinct positive integers. We denote by .f .S // D f .x i ; x j / and .f OES / D f OEx i ; x j the n n matrices having f evaluated at the greatest common divisor .x i ; x j / and the least common multiple OEx i ; x j of x i and x j as their .i; j /-entries, respectively. In 1875, Smith [25] published his famous result stating that det.f .S // D n Q iD1 .f /.x i / if S is factor closed (i.e., d 2 S if x 2 S and d jx), where f is the Dirichlet convolution of f and the Möbius function . Since then this topic has received a lot of attention from many authors and particularly became extremely active in the past decades (see, for example, [1]- [7], [9]- [23] and [26]- [28]).
In 1989, Beslin and Ligh [3] extended Smith's result by showing that det.S / D , .x i ; x j / 2 S for all integers i and j with 1 Ä i; j Ä n). In 1992, Bourque and Ligh [4] x/ , f is multiplicative and f .x/ ¤ 0 for all x 2 S. We say that S consists of multiple coprime gcd-closed sets if there is a positive integer h and h distinct gcdclosed sets S 1 ; :::; S h with .lcm.S i /; lcm.S j // D 1 for all integers i and j with 1 Ä i ¤ j Ä h such that S can be partitioned as the union of S 1 ; :::; S h (see, for instance, [15]). Clearly, if S consists of multiple coprime gcdclosed sets, then either we have 1 2 S or 1 6 2 S . For the former case 1 2 S, S is gcd closed and the formulas for determinants of the matrices .f .S // and .f OES / were given by Bourque and Ligh [5] and Hong [13], respectively. For the latter case 1 6 2 S, the formulas for determinants of the matrices .f .S // and .f OES / are unknown. This problem is still kept open so far.
In this paper, our main goal is to introduce a new method to investigate the above problem. Actually, we first give the formula for the determinant of .f .S // on any positive integers set S. Then we present formulas for the determinants of the matrices .f .S // and .f OES / on the multiple coprime gcd-closed sets S. Evidently, any rearrangement of the elements of S yields matrices similar to the matrices .f .S // and .f OES /. So we can rearrange the elements of S in any case of necessity. To give the main result, we need two concepts as follows.  For example, if S D f2; 5; 6; 8; 11; 35; 143g, then S consists of three coprime gcd-closed sets and the set M.S / of minimal elements of S is equal to f2; 5; 11g. Now we can state the main result of this paper. Then Furthermore, if f is a multiplicative function and f .x/ ¤ 0 for all x 2 S, then If letting S be a gcd-closed set, then Theorem 1.3 reduces to the Bourque-Ligh theorem [5] and Hong's theorem [13]. If S consists of coprime divisor chains, then Theorem 1.3 becomes the main result of [18]. From Theorem 1.3, one can easily deduce the following interesting consequence.
where I is the arithmetic function defined by I.n/ WD n.
Obviously, picking S to be a gcd-closed set in Corollary 1.4 gives us the Beslin-Ligh result [3] and the Bourque-Ligh result [4]. If S D M.S /, then Corollary 1.4 is Lemma 2.1 of [17]. We organize the paper as follows. In Section 2, we present some lemmas which are needed in the proof of Theorem 1.3. In Section 3, we prove Theorem 1.3 and Corollary 1.4.

Several lemmas
In this section, we present some useful lemmas that are needed in the next section. The first two lemmas are well known.
Proof. Clearly, the terms in the sum of the right-hand side of (2) are non-repetitive. Now we show that the terms in the sum of the left-hand side of (2) are non-repetitive. For this purpose, for any y 2 S with yjx, we let D.y/ D fd 2 Z C W d jy; d − z; z < y; z 2 S g. Claim that D.y 1 / T D.y 2 / D for any distinct elements y 1 and y 2 in the set S satisfying y 1 jx and y 2 jx. Otherwise, we may let d 2 D.y 1 / T D.y 2 /. Then d jy 1 and d jy 2 . So d j.y 1 ; y 2 /. But the assumption that S being gcd closed tells us that .y 1 ; y 2 / D y 3 for some y 3 2 S . Hence d jy 3 . On the other hand, we have y 3 < y 1 and y 3 < y 2 since y 1 ¤ y 2 . It then follows from d 2 D.y 1 / that d − y 3 . We arrive at a contradiction. The claim is proved. By the claim we know immediately that the terms in the sum of the left-hand side of (2) are non-repetitive.
For any term g.d / in the sum of the left-hand side of (2), one has d jy, yjx and y 2 S . Thus d jx. This implies that g.d / is a term in the sum of the right-hand side of (2). To show that the converse is true, for any given positive integer d and x 2 S with d jx, we let I.d; x/ D fu W d ju; ujx; u 2 Sg. Then I.d; x/ ¤ since x 2 I.d; x/ and I.d; x/ is finite. Let v D min.I.d; x//. Then vjx, v 2 S and d jv and d − z for any z 2 S with z < v. It infers that the term g.d / in the sum of the right-hand side of (2) is also a term in the sum of the left-hand side of (2). So (2) is proved.
This ends the proof of Lemma 2.3.
Note that a special case of Lemma 2.3 is due to Beslin and Ligh [3] and a more general form is given in (3.4) of [10]. We need the following definition to state Lemma 2.6 below. For 1 Ä l Ä m, we define E l .S / to be the n .m 1/ matrix obtained from E.S / by deleting its lth column.
We can now use the gcd-closed set to describe the structure of the matrix .f .S // on any set S of positive integers.
Li [21], Hong [12] and Mattila and Haukkanen [22] made use of the Cauchy-Binet formula to the Smith's matrices. Now we use this renowned formula to show the following lemma.
with E.S/ .k 1 ;:::;k n / being the n n matrix whose columns are the k 1 th, ..., k n th columns of E.S /. where A .k 1 ;:::;k n / is the n n matrix whose columns are the k 1 th, ..., k n th column of A. One can easily check that In what follows, we write S D S h iD1 S i with S i D fx i1 ; :::; x i n i g.1 Ä i Ä h/ being gcd closed and 1 < x i1 < ::: < x i n i and gcd.lcm.S i /; lcm.S j // D 1 for all integers i and j with 1 Ä i ¤ j Ä h. That is, S D fx 11 ; :::; x 1n 1 ; :::; x h1 ; :::; x hn h g: (6) Let N S WD S [ f1g D fx 11 ; :::; x 1n 1 ; :::; x h1 ; :::; x hn h ; 1g: Clearly N S is the minimal gcd-closed set containing S .

Proof. Let
Lemma 2.8. Let S be as in (6) and t be a given integer such that 1 Ä t Ä h. Let l t D n 1 C ::: C n t . Let n t 2. Then each of the following is true.
(i) If x t;n t 1 does not divide x t n t , then det.E l t .S // D det.E l t 1 .S n fx t;n t 1 g//: (ii) If x t;n t 1 divides x t n t , then det.E l t .S // D det.E l t 1 .S n fx t;n t 1 g// det.E l t 1 .S n fx t;n t g//: Proof. Since S is as in (6), by the definition of E.S / we have where for 1 Ä l Ä h, one has But E l t .S / is the l h l h matrix obtained from E.S / by deleting its l t th column. So one has (i). x t;n t 1 − x tn t . Then one has that e 0 n t ;n t 1 D 0. Thus the .l t 1/th column of E l t .S / is .0; :::; 0 "ƒ‚… l t 2 ; 1; 0; :::; 0 "ƒ‚… Then using the Laplace expansion theorem, we obtain that det. with On the other hand, by the definition of S, one can easily deduce that S n fx t;n t 1 g consists of multiple coprime gcd-closed sets and N S n fx t;n t 1 g is the minimal gcd-closed set containing the set S n fx t;n t 1 g. Hence by the definition of E l t 1 .S n fx t;n t 1 g/, one knows that the right-hand side of (8) is equal to det.E l t 1 .S n fx t;n t 1 g//. So the desired result follows. Part (i) is proved.  Clearly S n fx t;n t g consists of multiple coprime gcd-closed sets and N S n fx t;n t g is the minimal gcd-closed set containing S n fx t;n t g. Thus by the definition of E l t 1 .S n fx t;n t g/, we know that the right-hand side of (9) is equal to det.E l t 1 .S n fx t;n t 1 g// det.E l t 1 .S n fx t;n t g//: So part (ii) is true.
This concludes the proof of Lemma 2.8.
In ending this section, we show the following relation between S .1/ f .T / and S .2/ f .T / which is also needed in the proof of Theorem 1.3. Lemma 2.9. Let f be an arithmetic function and T be a set of distinct positive integers. If f .x/ ¤ 0 for any x 2 T and f .1/ D 1, then one has that S .1/ Proof. Since f .x/ ¤ 0 for any x 2 T and f .1/ D 1, it follows that f .x/ D S .2/ f .T / as desired. So Lemma 2.9 is proved.

Proofs of Theorem 1.and Corollary 1.4
In this section, we prove Theorem 1.3 and Corollary 1.4. We begin with the proof of Theorem 1.3.