NOTES ON THE DIVISIBILITY OF GCD AND LCM MATRICES

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1 NOTES ON THE DIVISIBILITY OF GCD AND LCM MATRICES PENTTI HAUKKANEN AND ISMO KORKEE Receive 10 November 2004 Let S {,,...,x n } be a set of positive integers, an let f be an arithmetical function. The matrices S f [ f gcx i,x j ] an [S] f [ f lcm[x i,x j ]] are referre to as the greatest common ivisor GCD an the least common multiple LCM matrices on S with respect to f, respectively. In this paper, we assume that the elements of the matrices S f an [S] f are integers an stuy the ivisibility of GCD an LCM matrices an their unitary analogues in the ring M n Zofthen n matrices over the integers. 1. Introuction Let S {,,...,x n } be a set of positive integers with < < <x n,anlet f be an arithmetical function. Let S f enote the n n matrix having f evaluate at the greatest commonivisorx i,x j ofx i an x j as its ij entry, that is, S f [ f x i,x j ]. Analogously, let [S] f enote the n n matrix having f evaluate at the least common multiple [x i,x j ]ofx i an x j as its ij entry, that is, [S] f [ f [x i,x j ]]. The matrices S f an [S] f arereferretoasthegcdanlcmmatricesons with respect to f, respectively. If f m m for all positive integers m, we enote S f S an[s] f [S]. Smith [16] calculate ets f when S is a factor-close set an et[s] f in a more special case. Since Smith, a large number of results on GCD an LCM matrices have been presente in the literature. For general accounts, see, for example, [7, 12]. In this paper, we assume that the elements of the matrices S f an [S] f are integers an stuy the ivisibility of GCD an LCM matrices in the ring M n Z ofthen n matrices over the integers. This stuy was begun by Bourque an Ligh [2, 4], who showe that i if S is a factor-close set, then S [S], see [2, Theorem 3], an, more generally, ii if S is a factor-close set an f is a multiplicative function such that f x i an f µx i are nonzero for all x i S,thenS f [S] f,see[4,theorem4]. Hong [8, 9, 10] has stuie the ivisibility of GCD an LCM matrices extensively. We review these results here: iii if n 3, then for any gc-close set S with n elements, S [S], see [8,Theorem 3.1i], Copyright 2005 International Mathematics an Mathematical Sciences 2005: DOI: /IJMMS

2 926 Notes on the ivisibility of GCD an LCM matrices iv for each n 4 there exists a gc-close set S with n elements such that S [S], see [8, Theorem 3.1ii], v for each n 4 there exists a gc-close set S with n elements such that ets et[s] in the ring of integers, see [9, Theorem 3.3ii]. Note that iv is a consequence of v, vi if S is a gc-close set such that each member of S is less than 12, then ets et[s],see [9, Theorem 3.5], vii if S is a multiple-close set an if f is a completely multiplicative function satisfying certain conitions or if S is a ivisor chain of positive integers an f satisfies a ivisibility conition, then S f [S] f,see[10, Theorems 4.5 an 5.1]. In this paper, we present some generalizations an analogues of the statements i v. Our results involve GCD, LCM, GCUD, an LCUM matrices, where GCUD stans for the greatest common unitary ivisor an LCUM stans for the least common unitary multiple. The number-theoretic concepts use in the introuction are explaine in Section Preliminaries In this section, we review the basic results on arithmetical functions neee in this paper. For more comprehensive treatments of arithmetical functions, we refer to [1, 13, 15]. The Dirichlet convolution f g of two arithmetical functions f an g is efine as f gn n n f g. 2.1 The ientity uner the Dirichlet convolution is the arithmetical function δ efine as δ1 1anδn 0forn 1. An arithmetical function f possesses a Dirichlet inverse f 1 if an only if f 1 0. Let ζ enote the arithmetical function efine as ζn 1for all n Z +.TheMöbius function µ is the Dirichlet inverse of ζ. The ivisor functions σ k are efine as σ k n n k for all n Z +. A ivisor of n is sai to be a unitary ivisor of n an is enote by n if,n/ 1. The unitary convolution of arithmetical functions f an g is efine as f gn n n f g. 2.2 The ientity uner the unitary convolution is again the arithmetical function δ.anarithmetical function f possesses a unitary inverse if an only if f 1 0. We enote the inverse of ζ uner the unitary convolution as µ. The function µ is referre to as the unitary analogue of the Möbius function. An arithmetical function f is sai to be multiplicative if f 1 1an f mn f m f n 2.3

3 P. Haukkanen an I. Korkee 927 whenever m,n 1, an an arithmetical function f is sai to be completely multiplicative if f 1 1an2.3holsforallm an n. An arithmetical function f is multiplicative if an only if f 1 1an f n f p np 2.4 p P for all n>1, where n p P p np is the canonical factorization of n.herep is the set of all prime numbers. For example, the Möbius function µ an its unitary analogue µ are multiplicative functions. The Dirichlet inverse of a completely multiplicative function f is given as f 1 µf. Likewise, the unitary inverse of a multiplicative function f is µ f but we o not nee this result here. An arithmetical function f is sai to be semimultiplicative if f m,n f [m,n] f m f n 2.5 for all m an n. See[12, 14, 15]. Multiplicative functions f are semimultiplicative functions f with f 1 1. An arithmetical function f is sai to be a totient if there exist completely multiplicative functions f t an f v such that f f t f 1 v ft µf v. 2.6 The functions f t an f v are referre to as the integral an inverse parts of f, respectively. Euler s φ-function is a famous example of a totient. It is well known that φ t N an φ v ζ,wherenn n for all n Z +.Deekin sψ-function efine as ψn p n1 + 1/p is another example of a totient. It is easy to see that ψ t N an ψ v λ, whereλ is Liouville s function see, e.g., [13]. Each completely multiplicative function f is a totient with f t f an f v δ, an each totient is a multiplicative function. In Theorem 3.4,we consier semimultiplicative functions f satisfying x i x j x i x j 2.7 an f x i Z \{0} for all x i,x j S. Integer-value totients f are examples of semimultiplicative functions satisfying 2.7forallx i,x j Z +,see[6,corollary3]. We enote the greatest common unitary ivisor gcu of m an n as m,n.the least common unitary multiple lcum of m an n, writtenas[m,n],isefineasthe least positive integer x such that m x an n x. It is easy to see that m,n exists for all m an n,an[m,n] exists if an only if for all prime numbers p,wehavemp np, mp 0, or np 0. If [m,n] exists, then [m,n] [m,n]anm,n m,n. The n n matrix having f evaluate at the gcu x i,x j of x i an x j as its ij entry is enote by S f, an the n n matrix having f evaluate at the lcum [x i,x j ] of x i an x j as its ij entry is enote by [S] f provie that [x i,x j ] exists for all x i an x j. The matrices S f an [S] f are referre to as the GCUD an LCUM matrices on S with respect to f, respectively. If f m m for all positive integers m, we enote S f an [S] f [S]. S

4 928 Notes on the ivisibility of GCD an LCM matrices The concepts of a factor-close, a gc-close, an lcm-close, a unitary ivisor-close, a gcu-close, an an lcum-close set are evient. The set S is sai to be multiple-close if S is lcm-close an if x i x n S hols for all x i S. We nee the following results on GCD an relate matrices. Bourque an Ligh [3, Corollary 1] show that if S is a factor-close set an f is an arithmetical function such that f µx i 0forallx i S,thenS f is invertible an S 1 f [a ij ], where a ij xi x j x k µ x k /x i µ /x j f µ. 2.8 It follows from [5, Theorem 6] that if S is a unitary ivisor-close set an f is an arithmetical function such that f µx i 0forallx i S, thens f is invertible an S f 1 [b ij ], where 3. Results b ij xi x j x k µ x k /x i µ x k /x j f µ x k. 2.9 In this section, we consier the ivisibility of GCD, LCM, GCUD, an LCUM matrices in the ring M n Z ofthen n matrices over the integers an the ivisibility of their eterminants in the ring of integers. Therefore, we assume that f x i,x j, f [x i,x j ], f x i,x j, an f [x i,x j ] are integers for all x i,x j S. In Theorem 3.1, we note that in the statement ii one nee not assume that f x i 0 for all x i S,aninTheorem 3.2, we propose a unitary analogue of ii. Theorem 3.1. Suppose that S is a factor-close set an f is a multiplicative function such that f µx i 0 for all x i S. Then S f [S] f. Proof. From 2.8, we see that the ij element of the matrix [S] f S 1 f is [S] f S 1 f ij n f [ ] µ x k /x m µ /x j x i,x m m1 xm x k f µ x j x k µ /x j f [ x i, ] µ x j x k f µ x k. 3.1 We show that f µ x k f [ xi, ] µ 3.2

5 for all k 1,2,...,n in the ring of integers. From 2.4, we obtain P. Haukkanen an I. Korkee 929 x k f [ xi, ] µ f p max{xip,p} µ p p p x k p P x kp p x k v0 f p max{xip,v} µ p p v p x i p x k f p x ip p x k f p max{x ip,x kp} f p max{xip,p 1} p x i p x k f p x ip f p x kp f p p 1 f p x ip, if p x k : x k p >x i p, p x k p x i p x k 0, if p x k : x k p x i p. 3.3 Thus x k f [ xi, ] µ xi f µ, if p x k : x k p >x i p,,x i 0, if p x k : x k p x i p. 3.4 Thus 3.2 hols. This shows that [S] f S 1 f M n Z. Theorem 3.2. Suppose that S is a unitary ivisor-close set such that [x i,x j ] exists for all i, j 1,2,...,n an suppose that f is a multiplicative function such that f µ x i 0 for all x i S. Then S f [S] f. Proof. From 2.9, we see that the ij element of the matrix [S] f S f 1 is [S] f S 1 f n ij m1 f [ x i,x m ] xm x k x j x k µ x k /x m µ x k /x j f µ x k µ /x j f [ x x j x k f µ i, ] µ x k x k. 3.5 We show that f µ x k x k f [ xi, ] µ 3.6

6 930 Notes on the ivisibility of GCD an LCM matrices for all k 1,2,...,n in the ring of integers. From 2.4, we obtain x k f [ xi, ] µ f p max{xip,p} µ p p p x k p p x i p P f p x ip f p xip f p x kp f 1 p x k p x i 0, if p : p x k p x i, f µ x k f xi, otherwise. p x k p x i f p x ip 3.7 Thus 3.6 hols. This shows that [S] f S f 1 M n Z. Remark 3.3. If [x i,x j ] exists as assume in Theorem 3.2, then[x i,x j ] [x i,x j ]an x i,x j x i,x j. However, the concepts of a factor-close set an a unitary ivisorclose set o not coincie. Thus Theorem 3.2 is not a special case of Theorem 3.1. In Theorem 3.4, we present a generalization an an lcm analogue of the statement iii in the introuction. If f m m for all m Z + an S is gc-close, then Theorem 3.4 reuces to the statement iii. In Remark 3.5, Theorem 3.6, anremark 3.7, wepropose unitary analogues of iii. Theorem 3.4. Let S be a gc-close or an lcm-close set with n elements, where n 3. Let f be a semimultiplicative function satisfying 2.7an f x i 0 for all x i,x j S. Then S f [S] f. Proof. Suppose first that S is a gc-close set with n elements. If n 1, then S f [S] f. Let n 2. Then an thus accoring to 2.7wehave f f an further [S] f S 1 f [ f x1 ][ f x1 ] f M 3 Z Let n 3. Then either or,.let.thenaccoringto2.7 we have f f f an further [S] f S 1 f f x f M 3 Z. x 3 f

7 P. Haukkanen an I. Korkee 931 Let,. Then, applying 2.5, we obtain f [, ] f f /f anapplying 2.7, we obtain f f, f. Thus [S] f S 1 f f f f f f x3 f f f f x3 f f f M 3 Z. 0 f Suppose secon that S is an lcm-close set with n elements. The cases n 1ann 2 are exactly the same as for a gc-close set. Let n 3. Then either or [, ]. The case is again exactly the same as for a gc-close set. Let [, ]. Then,applying2.5, we obtain f, f f /f an applying 2.7, we obtain f, f f. Thus [S] f S 1 f f f f f f x2 f f x 3 f x2 f M 3 Z Remark 3.5. It follows from Remark 3.3 an Theorem 3.4 that if S is a gcu-close or an lcum-close set with less than or equal to 3 elements, [x i,x j ] exists for all x i,x j S, an f is a semimultiplicative function satisfying 2.7an f x i 0forallx i,x j S,then S [S]. Theorem 3.6. Suppose that S is a gcu-close set with n elements, where n 3, anthat [x i,x j ] exists for all x i,x j S. Then ets et[s] i.e., ets et[s].

8 932 Notes on the ivisibility of GCD an LCM matrices Proof. If n 1, then S [S].Ifn 2, then an further ets anet[s]. Since,wehave a ± a for all a an, in particular, ets et[s]. Suppose that n 3. Then either or,.if,thenets x2 2 + anet[s] x Since,we have ets et[s].if,,thenets x1 1 + / an et[s] + /. Since,,wehavex1 2 an further ets et[s].fromremark 3.3,weseethatS San[S] [S], an therefore ets etsanet[s] et[s]. Remark 3.7. There exist lcum-close i.e., lcm-close sets S such that n 3, [x i,x j ] exists for all i, j an ets et[s] i.e., ets et[s]. For example, if S {2,3,6}, then ets et[s]. In Theorem 3.8, we present unitary an lcm analogues of statements iv an v in the introuction. Theorem 3.8. For each n 4, there exist a an lcum-close set S with n elements such that ets et[s] an so S [S], b a gcu-close set S with n elements such that ets et[s] an so S [S], c an lcm-close set S with n elements such that ets et[s] an so S [S], a gc-close set S with n elements such that ets et[s] an so S [S]. Proof. We first prove a. Let S {x 0,,,...,x n }, n 3, where x 0 1, p 1 p 2, p 1 p 3, x i p 1 p 2 p i for i 3,4,...,n. Herep 1, p 2,..., p n are some istinct prime numbers in increasing orer. It is clear that S is lcum-close. Then By row reuction, we obtain p 1 p 2 p 1 p 1 p 2 p 1 p 2 S 1 p 1 p 1 p 3 p 1 p 3 p 1 p 3 1 p 1 p 2 p 1 p 3, p 1 p 2 p 1 p 3 x n 1 p 1 p 2 p 1 p 3 x n p 1 p 2 p 1 p 2 x n [S] p 1 p 3 p 1 p 3 x n x n x n x n x n x n x n 3.12 ets eta 4 [ x3 p4 1 x n 1 pn 1 ], 3.13

9 where A 4 is the leaing principal 4 4submatrixofS, an thus P. Haukkanen an I. Korkee 933 ets p 2 1 p2 1 p p 3 p 2 + p 1 p 2 p 3 [ x3 p4 1 x n 1 pn 1 ] Similarly, et[s] etb 4 [ x4 1 p4 xn 1 pn ], 3.15 where B 4 is the leaing principal 4 4submatrixof[S], an thus et[s] p1 3 p2 2 p3 2 p2 1 p 3 1 [ ] p 1 p 2 p 1 p 3 + p 1 p 2 p 3 x4 1 p4 xn 1 pn. If we let p 1 2, p 2 3, an p 3 5, then et[s] ets 1n 1 p 1 p2 2 p 2 [ ] 3 1 p1 p 2 p 1 p 3 + p 1 p 2 p 3 p4 p 5 p n 1 p 3 p 2 + p 1 p 2 p 3 1n p 4 p 5 p n Let p 4, p 5,..., p n 23. Then ets et[s] an so S [S]. Thus a hols. Next we prove b. Consier the set S {x 0,,,...,x n }, n 3, where x 0 1, p 1, p 2, x i p 1 p 2 p i for i 3,4,...,n. Herep 1, p 2,..., p n are some istinct prime numbers in increasing orer. Clearly, S is gcu-close. For the sake of brevity, we o not present the matrices S an [S] explicitly. By row reuction, we obtain ets eta 4 [ x3 p4 1 x n 1 pn 1 ], 3.18 where A 4 is the leaing principal 4 4submatrixofS, an thus ets p 1 1 p p 1 p 2 + p 1 p 2 p 3 [ x3 p4 1 x n 1 pn 1 ] Similarly, et[s] etb 4 [ x4 1 p4 xn 1 pn ], 3.20 where B 4 is the leaing principal 4 4submatrixof[S], an thus et[s] p1 2 p2 2 p 3 p1 1 p 2 1 [ ] p 1 p 3 p 2 p 3 + p 1 p 2 p 3 x4 1 p4 xn 1 pn.

10 934 Notes on the ivisibility of GCD an LCM matrices If we let p 1 2, p 2 3anp 3 5, then et[s] ets 1n 1 p1 2 p 2 2 p 3 1 p1 p 3 p 2 p 3 + p 1 p 2 p 3 p4 p 5 p n 1 p 1 p 2 + p 1 p 2 p 3 1n p 4 p 5 p n Let p 4, p 5,..., p n 13. Then ets et[s] an so S [S]. Thus b hols. Since S in a is also lcm-close an since S S an [S] [S],wehaveetS et[s] ansos [S]. Thus c hols. Since S in b is also gc-close an since S S an [S] [S],wehaveetS et[s]ansos [S]. Thus hols. Next we present some minor notes on the statements ii, iv, v, an vii in the introuction. The statement ii oes not hol in general if f is not a multiplicative function. For example, if f 1 2, f 2 1, an S {1,2}, then f is not a multiplicative function, S is a factor-close set, ets f et[s] f an S f [S] f.thechoice f 1 2, f 2 1, f 3 4, an S {1,2,3} is an example such that f is not a multiplicative function, S is a factor-close set, ets f et[s] f but S f [S] f. Further, the statement ii oes not hol in general if S is a gc-close set, that is, not factor-close. The statement iv gives counterexamples for each n 4.Wecanalsofin counterexamples for n 2ann 3. In fact, for n 2let f be a multiplicative function such that f 2 2an f 4 1anletSbe the gc-close set given as S {2,4}. Then ets f et[s] f an so S f [S] f.forn 3let f be a multiplicative function such that f 2 2, f 4 1, an f 8 1anletSbe the gc-close set given as S {2,4,8}.Then ets f et[s] f an so S f [S] f.if f is a multiplicative function such that f 2 2, f 4 1, an f 8 2anifS is again the gc-close set given as S {2,4,8}, then ets f et[s] f but S f [S] f. In the statements iv an v, we note that there exist gc-close sets S such that ets et[s] buts [S], for example, S {1,2,3,12}. Similarly, S {1,4,6,12} is an example of an lcm-close set such that ets et[s]buts [S]. In the statement vii, Hong [10] notes that there exist multiplicative functions f an multiple close sets S such that S f [S] f,forexample, f σ 1 an S {6,8,12,24}. A more simple example is f σ 0 an S {2,4}. The pair f σ 0 an S {2,4,8} is an example such that ets f et[s] f but S f [S] f. Finally, we note that [11, Conjectures 5.3 an 5.4] o not hol. In fact, let k be a positive integer an let f be an arithmetical function efine as f n n k.letsbe a finite set of o positive integers. Conjectures 5.3 an 5.4 state that if S is gc-close or lcmclose, then S f [S] f.however,ifs {1,3,5,45}, thens is gc-close but S f [S] f. Namely, calculation with the Mathematica system shows that, for example, the 2, 4 entry of the matrix [S] f S 1 f is 1 3 k 5 k +15 k 15 k +45 k k 2 5 k +3 1+k 5 k +9 k +25 k 75 k 135 k 225 k k, 3.23

11 P. Haukkanen an I. Korkee 935 which is never an integer. Similarly, if S {1,9,15,45}, thens is lcm-close but S f [S] f. Again, calculation with the Mathematica system shows that, for example, the 2, 4 entry of the matrix [S] f S 1 f is 3 k 2 9 k +27 k 9 k +45 k 3 1+3k 5 k +9 k 2 27 k 2 45 k +81 k k 675 k 1215 k 2025 k k, 3.24 which is never an integer. The authors have alreay announce these two counterexamples {1,3,5,45} an {1,9,15,45} in review on [11] by P. Haukkanen. References [1] T.M.Apostol,Introuction to Analytic Number Theory, Unergrauate Texts in Mathematics, Springer, New York, [2] K. Bourque an S. Ligh, On GCD an LCM matrices, Linear Algebra Appl , [3], Matrices associate with arithmetical functions, Linear an Multilinear Algebra , no. 3 4, [4], Matrices associate with multiplicative functions, Linear Algebra Appl , [5] P. Haukkanen, On meet matrices on posets, Linear Algebra Appl , [6], Some characterizations of totients, Int. J. Math. Math. Sci , no. 2, [7] P. Haukkanen, J. Wang, an J. Sillanpää, On Smith s eterminant, Linear Algebra Appl , [8] S. Hong, On the factorization of LCM matrices on gc-close sets, Linear Algebra Appl , [9], Divisibility of eterminants of least common multiple matrices on GCD-close sets, Southeast Asian Bull. Math , no. 4, [10], Factorization of matrices associate with classes of arithmetical functions, Colloq. Math , no. 1, [11], Notes on power LCM matrices,actaarith , no. 2, [12] I. Korkee an P. Haukkanen, On meet an join matrices associate with incience functions, Linear Algebra Appl , [13] P. J. McCarthy, Introuction to Arithmetical Functions, Universitext, Springer, New York, [14] D. Rearick, Semi-multiplicative functions, Duke Math. J , [15] R. Sivaramakrishnan, Classical Theory of Arithmetic Functions, Monographs an Textbooks in Pure an Applie Mathematics, vol. 126, Marcel Dekker, New York, [16] H. J. S. Smith, On the value of a certain arithmetical eterminant, Proc. Lonon Math. Soc , Pentti Haukkanen: Department of Mathematics, Statistics an Philosophy, University of Tampere, Tampere, Finlan aress: pentti.haukkanen@uta.fi Ismo Korkee: Department of Mathematics, Statistics an Philosophy, University of Tampere, Tampere, Finlan aress: ismo.korkee@uta.fi

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