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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 14, Issue 1 (Jan 2016)

Issues

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

Siao Hong / Shuangnian Hu / Shaofang Hong
Published Online: 2016-03-19 | DOI: https://doi.org/10.1515/math-2016-0014

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, jn. 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 ≤ ijk). 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.

Keywords: Matrix associated with arithmetic function; Determinant; Multiple coprime gcd-closed sets; Smith’s determinant

MSC 2010: 11C20; 11A05; 15B36

1 Introduction and statements of main results

Let n be a positive integer and f be an arithmetic function. Let S = {x1, …, xn} be a set of n distinct positive integers. We denote by (f(S)) = (f(xi, xj)) and (f[S]) = (f[xi , xj]) the n × n matrices having f evaluated at the greatest common divisor (xi, xj) and the least common multiple [xi, xj] of xi and xj as their (i, j)-entries, respectively. In 1875, Smith 25 published his famous result stating that det(f(S))=i=1n(f*μ)(xi) if S is factor closed (i.e., dS if xS and d |x), where f * μ is the Dirichlet convolution of f and the Möbius function μ. 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)=i=1nd|xi,dxtxt<xiφ(d) if S is gcd closed (i.e., (xi, xj) ∈ S for all integers i and j with 1 ≤ i, jn). In 1992, Bourque and Ligh 4 proved that if S is gcd closed, then det[S]=i=1nxi2d|xi,dxtxt<xi g(d) with g being the multiplicative function defined by

g(m):=1md|mdμ(d).

In 1993, Bourque and Ligh 5 generalized Smith’s determinant and Beslin and Ligh’s result 3 by proving that if S is gcd closed, then det(f(S))=xSαS,f(x), where

αS,f(x):=d|x,dyy<x,yS(fμ)(d).(1)

In 2002, Hong 13 extended the Bourque-Ligh result by showing that det(f[S])=i=1nf(xi)2αS,1f(xi) if S is gcd closed, where 1f(x):=1f(x), f is multiplicative and f(x) ≠ 0 for all xS.

We say that S consists of multiple coprime gcd-closed sets if there is a positive integer h and h distinct gcd-closed sets S1,…, Sh with (lcm(Si), lcm(Sj)) = 1 for all integers i and j with 1 ≤ ijh such that S can be partitioned as the union of S1, …, Sh (see, for instance, 15). Clearly, if S consists of multiple coprime gcd-closed sets, then either we have 1 ∈ S or 1 ∉ S. For the former case 1 ∈ S, S is gcd closed and the formulas for determinants of the matrices (f(S)) and (f[S]) were given by Bourque and Ligh 5 and Hong 13, respectively. For the latter case 1 ∉ S, the formulas for determinants of the matrices (f(S)) and (f[S]) 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[S]) 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[S]). So we can rearrange the elements of S in any case of necessity. To give the main result, we need two concepts as follows.

Definition 1.1. Let f be an arithmetic function and T be a set of distinct positive integers. Then associated to f, we define the functions sf(1) and sf(2) on the set T as follows: sf(1)(T):=f(1)yTzT\{y}(f(z)f(1))+yT(f(y)f(1)),sf(2)(T):=(1)|T|+1(yTzT\{y}(f(z)1)+(|T|1)yT(f(y)1))yTf(y).

Definition 1.2. Let S consist of h coprime gcd-closed sets S1, …, Sh. Then we define the set of minimal elements of S, denoted by M(S), to be M(S):={min(Si)}i=1h,, where min(Si) stands for the smallest element of Si.

For example, if S ={2, 5, 6, 8, 11, 35, 143}, then S consists of three coprime gcd-closed sets and the set M(S) of minimal elements of S is equal to {2, 5, 11}.

Now we can state the main result of this paper.

Theorem 1.3. Let f be an arithmetic function. Let S consist of multiple coprime gcd-closed sets such that 1 ∉ S and M(S) denote the set of minimal elements of S. Then

det(f(S))=sf(1)(M(S))xS\M(S)αS,f(x).

Furthermore , if f is a multiplicative function and f(x) ≠ 0 for all x ∈ S, then

det(f[S])=sf(2)(M(S))xS\M(S)f(x)2αS,1f(x).

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.

Corollary 1.4. Let S consist of multiple coprime gcd-closed sets and M(S) denote the set of minimal elements of S. Then

det((S))=sI(1)(M(S))xS\M(S)d|x,dyy<x,ySφ(d)

and

det([S])=sI(2)(M(S))xS\M(S)x2d|xi,dxtxt<xig(d),

where I is the arithmetic function defined by I (n) := 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 = 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.

2 Several lemmas

In this section, we present some useful lemmas that are needed in the next section. The first two lemmas are well known.

Lemma 2.1 (13). Let f be any arithmetic function and n be a positive integer. Then d|n(f*μ)(d)=f(n)..

Lemma 2.2 (24). Let m , n be any positive integers and f be a multiplicative function. Then f(m)f(n) = f((m, n))f([m, n]).

Lemma 2.3. Let g be any arithmetic function and S be gcd closed. Then for any xS, we have

y|xySd|y,dzz<y,zSg(d)=d|xg(d).(2)

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 yS with y | x, we let D(y) = {dZ+ : d | y, dz, z < y, zS}. Claim that D(y1) ∩ D(y2) = ϕ for any distinct elements y1 and y2 in the set S satisfying y1| x and y2| x. Otherwise, we may let dD(y1) ∩ D(y2) . Then d | y1 and d | y2. So d |(y1, y2). But the assumption that S being gcd closed tells us that (y1, y2) = y3 for some y3S. Hence d | y3. On the other hand, we have y3 < y1 and y3 < y2 since y1y2. It then follows from dD(y1) that dy3. 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 | y, y| x and yS. Thus d | x. 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 xS with d | x, we let I(d, x) = {u : d| u, u| x, uS}. Then I(d, x) ≠ ϕ since xI(d, x) and I(d, x) is finite. Let υ = min(I(d, x)). Then υ | x, υS and d |υ and dz for any zS with z < υ. 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. □

Lemma 2.4. Let S be gcd closed. Then for any xS, y|xySαS,f(y)=f(x).

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.

Proof. Letting g = f * μ in Lemma 2.3 gives us that

y|xySd|y,dzz<y,zS(f*μ)(d)=d|x(f*μ)(d).

Then the desired result follows from the definition of αS, f (d) and Lemma 2.1. This completes the proof of Lemma 2.4. □

We need the following definition to state Lemma 2.6 below.

Definition 2.5. Let S = {x1, …, xn} be a set of positive integers and S = {y1, …, ym} be the minimal gcd-closed set containing S. Then we define the n × m matrix E(S) = (eij) by

eij:={1,if  yj|xi,0,  otherwise.

For 1 ≤ l ≤ m , we define El (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.

Lemma 2.6. Let f be an arithmetic function and S = {xi, …., xn} be a set of distinct positive integers and S = {y1, …. , ym} be the minimal gcd-closed set containing S. Then (f(S)) = E(S) · diag(αS, f (y1), …, αS, f (ym)) · E(S)T.

Proof. Let S = {x1, …, xn} and ∆ = diag(αS, f(y1), , αS, f (ym)). Then for any integers i and j with 1 ≤ i, jn, we have

(E(S)ΔE(S)T)ij=k=1meikαS¯,f(yk)ejk=yk|xiyk|xjαS¯,f(yk)=yk|(xi,xj)αS¯,f(yk).

Since S is the minimal gcd-closed set containing S, one has (xi, xj) ∈ S. Then there exists one element yhS such that yh = (xi, xj). It follows that

(E(S)ΔE(S)T)ij=yk|yhαS,¯f(yk)(3)

But Lemma 2.4 together with the fact that S being gcd closed implies that

yk|yhαS¯,f(yk)=f(yh)=f(xi,xj).(4)

Thus by (3) and (4), one has (E(S)∆E(S)T)ij = (f(S))ij as desired. This completes the proof of Lemma 2.6. □

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.

Lemma 2.7. Let f be an arithmetic function and S = {x1, …., xn} be a set of n distinct positive integers , and S = {y1, …, ym} be the minimal gcd-closed set containing S. Then

det(f(S))=1k1<<knm(detE(S)(k1,...,kn))2i=1nαS¯,f(yki),(5)

with E(S)(k1, …, kn) being the n × n matrix whose columns are the k1th, … , knth columns of E(S).

Proof. Let A = E(S) · diag(αS¯,f(y1),,αS¯,f(ym)). Then by Lemma 2.6, one has (f(S)) = AAT. Using the Cauchy-Binet formula 8 we get

det(f(S))=1k1knmdetΑ(k1,,kn)detΑT(k1,,kn)=1k1knm(detΑ(k1,,kn))2,

where A(k1, …, kn) is the n × n matrix whose columns are the k1th, …, knth column of A. One can easily check that

Α(k1,,kn)=E(S)(k1,,kn)diag(αS¯,f(yk1),,αS¯,f(ykn)).

It then follows that

det(Α(k1,,kn))=i=1nαS¯,f(yki)det(E(S)(k1,,kn)).

So the desired formula (5) follows immediately. This finishes the proof of Lemma 2.7. □

In what follows, we write S=i=1hSi with Si = {xi1, …, xini }(1 ≤ ih) being gcd closed and 1 < xi1 < … < xini and gcd(lcm(Si), lcm(Sj)) = 1 for all integers i and j with 1 ≤ ijh. That is,

S=  {x11,,x1n1,,xh1,,xhnh}.(6)

Let S: = S ∪{1} = {x11,…, x1n1,…, xh1,…, xhnh, 1}. Clearly S is the minimal gcd-closed set containing S.

Lemma 2.8. Let S be as in (6) and t be a given integer such that 1 ≤ th. Let lt = n1 + … + nt. Let nt ≥ 2. Then each of the following is true.

(i) If xt , nt –1 does not divide xtnt, then det(Elt (S)) = det(Elt1(S \ {xt,nt –1})).

(ii) If xt nt –1 divides xtnt, then

det(Elt(S))=det(Elt1(S\{xt,nt1}))det(Elt1(S\{xt,nt})).

Proof. Since S is as in (6), by the definition of E(S) we have

E(S)=(E10010E20100Eh1),(7)

where for 1 ≤ lh, one has

El=(10000110001e321001enl1,2enl1,3101enl,2enl,3enl,nl11)

with eij (1 ≤ i, jnt) being defined as

eij={1,  if  xlj|xli,0,  otherwise.

But Elt(S) is the lh × lh matrix obtained from E(S) by deleting its ltth column. So one has

Elt(S)=(E10010Et0100010Eh1)

where

Et=(100011001e32101ent1,2ent1,311ent1,2ent,3ent,nt1).

(i). xt, nt–1. ∤ xtnt. Then one has that ent,nt1=0. Thus the (lt – 1)th column of Elt(S) is (0,,0,lt21,0,,0,)Tlhlt+1. Then using the Laplace expansion theorem, we obtain that

det(Elt(S))=det((E10010Et01000100Eh1)),(8)

with Et=(100011001e32101ent2,2ent2,311ent,2ent,3ent,nt2).

On the other hand, by the definition of S, one can easily deduce that S \ {xt, nt –1} consists of multiple coprime gcd-closed sets and S \ {xt, nt –1} is the minimal gcd-closed set containing the set S \ {xt, nt –1}. Hence by the definition of Elt – 1(S \ {xt, nt –1}), one knows that the right-hand side of (8) is equal to det(Elt – 1(S \ {xt, nt –1})). So the desired result follows. Part (i) is proved.

(ii). xt, nt–1. | xtnt. Thus ent,nt1=1. Clearly the (lt – 1)th column of Elt(S) is (0,,0,lt21,1,0,,0,)Tlhlt. Applying the Laplace expansion theorem gives us that

det(Elt(S))=det((E10010Et01000100Eh1))det((E10010Et01000100Eh1))(9)

With

Et=(100011001e32101ent2,2ent2,311ent1,2ent1,3ent1,nt2).

Clearly S \ {xt, nt} consists of multiple coprime gcd-closed sets and S \ {xt, nt} is the minimal gcd-closed set containing S \ {xt, nt}. Thus by the definition of Elt–1(S \ {xt, nt}), we know that the right-hand side of (9) is equal to det(Elt–1(S \ {xt,nt–1})) – det(Elt–1(S \ {xt, nt})) So part (ii) is true.

This concludes the proof of Lemma 2.8. □

In ending this section, we show the following relation between Sf(1)(T) and Sf(2)(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 xT and f(x) = 1 then one has that S1f(1)(T)xTf(x)2=Sf(2)(T).

Proof. Since f(x) ≠ 0 for any xT and f(x) = 1, it follows that

S1f(1)(T)xTf(x)2=(yTxT\{y}(1f(x)1)+yT(1f(y)1))xTf(x)2=(yTxT\{y}1f(x)f(x)+yT1f(y)f(y))xTf(x)2=(yTf(y)xT\{y}(1f(x))+yT(1f(y)))xTf(x)=(1)|T|+1(yT((f(y)1)+1)xT\{y}(f(x)1)+(1)|T|yT(f(y)1))xTf(x)=(1)|T|+1(yTxT(f(x)1)+yTxT\{y}(f(x)1)yT(f(y)1)xTf(x)=(1)|T|+1((xT(f(x)1)))yT1+yTxT\{y}(f(x)1)yT(f(y)1)xTf(x)=(1)|T|+1(yTxT\{y}(f(x)1)+(|T|1)yT(f(y)1)xTf(x)=Sf(2)(T)

as desired. So Lemma 2.9 is proved. □

3 Proofs of Theorem 1.3 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.

Proof of Theorem 1.3. Since S consists of multiple coprime gcd-closed sets such that 1 ∉ S, one may write S as in the form of (6). For 1 ≤ ih , let li = n1 + … + ni. Then the lh × (lh + 1) matrix E(S) is of the form (7).

Let’s first deal with det(f(S)). Define x1 = x11, …, xl1 = x1n1, …, xlh – 1 + 1 = xh1, …, xlh = xhnh, xlh + 1 = 1 and l0 = 0. Lemma 2.7 tells us that

det(f(S))=1k1<<k1hlh+1(det(E(S)(k1,,k1h)))2i=1lhαS¯,f(xki).(10)

For any {k1,…, klh} with 1 k1 < … < kh < lh + 1, write {k1,…, klh} := {1,2,…, lh + 1 } \ {k}. Then det(E(S)(k1,…, kth)) = det(Ek (S)).

We claim that (det(E(S)(k1 ,…, kt)))2 = 1 if k = li + 1 for some i ∈ {0, 1, …, h}, and det(E(S)(k1,…,klh)) = 0 if k{li+1}i=0h.

First, let k = li + 1 for some integer i with 0 ≤ ih. If 0 ≤ ih 1, then the li + 1 row of Eli + 1(S) is (0,…, 0,1). So the Laplace expansion applied to det(Eli + 1(S)) gives us a lower triangular determinant with all the diagonal elements being 1. It follows that

(det(E(S)(k1,,klh)))2=(det(Eli+1(S)))2=1.

If i = h, then Elh + 1(S) is a lower triangular matrix with all the diagonal elements being 1. Hence det (E(S)(k1),…,klh))= det(Elh + 1(S)) = 1. Therefore the first part of the claim is true.

Obviously, the second part of the claim is equivalent to the statement that det(Et (S)) = 0 for all integers t withli–1 + 2 ≤ tli and 1 ≤ih, which will be proved in the following.

Given any integer i with 1 ≤ih. We prove the claim by using induction on ni. If ni = 2, then t = li = li–1 + 2. Further,xi1xi2. Using Lemma 2.8, wehavedet(Eli(S)) = det(Eli + 1(S\{xi1}))–det(Eli –1(S)\{xi2})). Since ni = 2, we derive that Eii – 1(S\{xi1}) = Eii – 1(S\{xi2}). This implies immediately that det(Eli (S)) = 0. So the claim is true ifni = 2.

Let ni ≥ 3 and assume that the claim is true for the ni – 1 case. In what follows we consider the ni case. For li–1 + 2 ≤ tli – 1, noting that all elements of the (li – 1)-th column of Et (S) are zero except for its (li,li – 1)-entry is 1, applying the Laplace theorem to det(Et (S)) gives that det(Et (S)) = – det(Et (S \ {xi,ni})). But the inductive assumption implies that det(Et (S \ {xi,ni})) = 0. Thus det(Et (S)) = 0 as claimed. Now lett =li .If xi,ni 1xini, then Lemma 2.8 infers that det(Eli. (S)) = det(Eli– 1 (S \ {xi,ni – 1})). However, by the induction assumption we have det(Eli – 1(S\{xi,ni – 1})) = 0. Hence det(Eli. (S)) = 0 as required. If xini–1 |xini, by Lemma 2.8 we have

det(Eli(S))=det(Eli1(S\{xi,ni1}))det(Eli1(S\{xi,ni})).

But the induction assumption tells us that

det(Eli1(S\{xi,ni1}))=det(Eli1(S\{xi,ni}))=0.

So det(Eli (S)) = 0 and the claim is proved.

Now, by (10) and the claim, one deduces that

det(f(S))=i=0h1j=1jli+1lh+1αS¯,f(xj)+i=1lhαS¯,f(xi).(11)

From (1), we deduce thatαS, f (1) =f(1) andαS, f (xi1) =f(xi1) –f(1). It then follows from (11) that

det(f(S))=f(1)(i=0h1j=1jli+1lhαS¯,f(xj))+i=1lhαS¯,f(xi)=(f(1)i=1hj=1jilhαS¯,f(xj1)+i=1hαS¯,f(xi1))i=1hj=2niαS,f(xij)=(f(1)i=1hj=1jih(f(xj1)f(1))+i=1h(f(xi1)f(1)))i=1hj=2niαS,f(xij)=Sf(1)(M(S))xS\M(S)αS,f(x)

as desired.

Finally, we turn our attention to det(f[S]). Letf be a multiplicative function andf(x) ≠ 0 for allxS. Thenf(1) = 1 and from Lemma 2.2 we derive that

(f[S])=diag(f(x1),,f(xlh))(1f(xi,xj))diag(f(x1),,f(xlh)).

Therefore det(f[S])=det(1f(S))i=1lhf(xi)2. Thus the formula for det(f(S)) applied to det(1f(S)) and Lemma 2.9 applied to the setT =M(S) give us that

det(f[S])=S1f(1)(M(S))(xS\M(S)αS,1f(x))xSf(x)2=S1f(1)(M(S))(xM(S)f(x)2)xS\M(S)f(x)2αS,1ff(x)=Sf(2)(M(S))xS\M(S)f(x)2αS,1f(x)

as required. This finishes the proof of Theorem 1.3. □

We are now ready to prove Corollary 1.4 as the conclusion of this paper.

Proof of Corollary 1.4. Letf =I. Then Theorem 1.3 tells us that

det((S))=sI(1)(M(S))xS\M(S)αS,Ι(x).

By the definition of αS,I (x) and I * μ = φ, we have

αS,I(x)=d|x,dyy<x,yS(Ιμ)(d)=d|x,dyy<x,ySφ(d).

Thus the desired result follows immediately. By Theorem 1.3, one has

det([S])=sI(2)(M(S))xS\M(S)x2αS,1I(x)=sI(2)(M(S))xS\M(S)x2d|x,dyy<x,yS(1Iμ)(d).

Since

(1Iμ)(d)=d|dddμ(d)=1dd|ddμ(d)=g(d),

the desired result then follows immediately. This completes the proof of Corollary 1.4. □

Remark 3.1.If S consists of multiple coprime gcd-closed sets such that 1 ∉ S, then Theorem 1.3 gives us formulas for det(f(S)) and det(f[S]). One can easily see that Theorem 1.3 is not true if S consists of multiple gcd-closed sets which are not coprime. If S consists of multiple gcd-closed sets such that gcd(S) = 1 ∉ S and these gcd-closed sets are not coprime, then what are the formulas for det(f(S)) and det(f[S])? This interesting problem keeps open.

Acknowledgement

This research was supported partially by National Science Foundation of China Grant # 11371260. The authors thank the anonymous referees for helpful comments and suggestions.

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About the article

Received: 2014-03-01

Accepted: 2014-08-26

Published Online: 2016-03-19

Published in Print: 2016-01-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0014.

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© 2016 Hong et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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