Hankel determinants with arithmetic functions

Hankel determinants with arithmetic functions Andrzej Nowicki Toru´ n 18.06.2017

Contents 1 Introduction

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2 Notations and preparatory facts

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3 Gcd-determinants 3.1 Smith’s theorem . . . . . . . . 3.2 Smith determinants . . . . . . 3.3 Power gcd-determinants . . . 3.4 Gcd-determinants with tau . . 3.5 Gcd-determinants with sigma 3.6 Other gcd-determinants . . .

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4 Lcm-determinants 4.1 Smith lcm-determinants . . . . . . . . 4.2 Power lcm-determinants . . . . . . . . 4.3 Examples of lcm-determinants . . . . . 4.4 Initial values of some lcm-determinants

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5 Determinants with some arithmetic functions

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6 Appendix 6.1 Arithmetic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Examples of factor-closed sets . . . . . . . . . . . . . . . . . . . . . . .

21 21 26

1

Introduction

A Hankel determinant, named after Hermann Hankel, is the determinant of a square matrix in which each ascending skew-diagonal from left to right is constant, e.g.: a b c d a b c b c d e b c d , c d e f . c d e d e f g In this article we consider some Hankel determinants. If a, b are positive integers, then we use the standard notations (a, b) = gcd(a, b) and [a, b] = lcm(a, b).

Andrzej Nowicki, 2017,

Hankel determinants with arithmetic functions

In 1875, J. H. S. Smith [32] proved (1, 1) (1, 2) · · · (2, 1) (2, 2) · · · .. .. . . (n, 1) (n, 2) · · · where ϕ is the Euler

2

that = ϕ(1)ϕ(2) · · · ϕ(n), (n, n) (1, n) (2, n) .. .

totient function. He also showed that [1, 1] [1, 2] · · · [1, n] Y [2, 1] [2, 2] · · · [2, n] [n/p] , .. .. .. = n! (1 − p) . . . p [n, 1] [n, 2] · · · [n, n]

where p ranges over all primes belonging to the set {1, 2, . . . , n}. Moreover, Smith presented several generalizations of the above determinants. Since Smith, a large number of papers on this topic has been published in the literature (see for example [1, 4, 5, 14, 18, 19, 22, 23, 30]). In this article we present our proofs of theorems concerning Smith’s determinants. Moreover, we present a collections of determinants with great common divisors, least common multiplies and some arithmetic functions.

2

Notations and preparatory facts

We denote by N the set {1, 2, . . . }, of all natural numbers. If a, b ∈ N, then we use the standard notations (a, b) = gcd(a, b) and [a, b] = lcm(a, b). A finite set S of positive integers is said to be factor-closed if all positive factors of any element of S belong to S, ([4, 22]). For any n ∈ N, the set S = {1, 2, . . . , n} is factor-closed. Other examples are in Appendix. A function f is said to be arithmetic, if f is an ordinary function from N to the field C of complex numbers. We denote by A the set of all arithmetic functions. If n ∈ N, then τ (n) is the number of all natural divisors of n, and σ(n) is the sum of all natural divisors of n. Moreover, ϕ is the Euler totient function, and µ is the Möbius function. We use also the standard functions I, T , T r and e. If n is a positive integer, then ( 1, for n = 1 1 = I(n) = 1, T (n) = n, T r (n) = nr , e(n) = n 0, for n > 1. If f, g : N → C are arithmetic functions, then we denote by f ∗ g the Dirichlet convolution of f and g, that is, f ∗ g is a function from N to C defined by n X f ∗ g (n) = f (d)g , d d|n

for all n ∈ N, where d ranges over all positive divisors of n. Basic properties of the Dirichlet convolution one can find in many papers and books (see for example



3

[11, 12, 26, 28]). It is well known that the set A (of all arithmetic functions) is a commutative ring with respect to the ordinary addition + and the Dirichlet convolution, as the multiplication. This ring is without zero divisors, and the function e is its identity. An arithmetic function f is invertible in this ring if and only if f (1) 6= 0. If a function f ∈ A is invertible in A, then we denote by f −1 the inverse of f , that is f ∗ f −1 = e. In this case we say that f −1 is the inverse of f with respect to the Dirichlet convolution. A multiplicative function is an arithmetic function f with the property f (1) = 1 and whenever a and b are coprime, then f (ab) = f (a)f (b). All the functions e, I, T , ϕ, τ , σ and µ are multiplicative. If f, g are multiplicative functions, then the ordinary product f g, n 7→ f (n)g(n), is also a multiplicative function. The following two propositions are well known ([2, 17, 25, 28, 33]). Proposition 2.1. The Dirichlet convolution of multiplicative functions is an multiplicative function. Proposition 2.2. If f is a multiplicative function, then f has the inverse f −1 (with respect to the Dirichlet convolution), and the function f −1 is multiplicative. Thus, the set of all multiplicative functions is a subgroup of the multiplicative group of the ring A. An arithmetic function f is said to be completely multiplicative (or totally multiplicative) if f (1) = 1 and f (ab) = f (a)f (b) holds for all positive integers a and b, even when they are not coprime. The functions e, I, T and T r are completely multiplicative. The Dirichlet convolution of completely multiplicative functions is not completely multiplicative, in general. For example, I is completely multiplicative, and τ = I ∗ I is not completely multiplicative. Note also that µ = I −1 , and hence the inverse of a completely multiplicative function is not, in general, completely multiplicative. If f is a completely multiplicative function, then f has the inverse with respect to Dirichlet convolution and the inverse is equal to µf (see Proposition 6.6 in Appendix). Note several important equalities. τ = I ∗ I, σ = T ∗ I, ϕ = T ∗ µ, T = ϕ ∗ I, µ ∗ σ = T, ϕ ∗ τ = σ. I −1 = µ, T −1 = µT, ϕ−1 = I ∗ µT = I ∗ T −1 , τ −1 = µ ∗ µ, σ −1 = T −1 ∗ µ = µT ∗ µ = ϕ−1 ∗ τ −1 . For any positive integer n and real r, we define Y 1 r Jr (n) = n 1− r , p p|n

where p ranges over all prime divisors of n. The function Jr is usually called Jordan’s totient. It is a multiplicative function. In particular, J1 = ϕ, J0 = e. It is easy to show (see Proposition 6.7) that Jr ∗ I = T r . This implies that Jr = T r ∗ µ,

Jr−1 = µT r ∗ I.



4

where Jr−1 is the inverse of Jr with respect to Dirichlet convolution. It is clear that [n/p] Y 1 r 1− r Jr (1)Jr (2) · · · Jr (n) = (n!) . p p For any real r, we denote by πr the multiplicative function defined by πr (pm ) = −pr , for each prime power pm . Thus, for n ∈ N, we have Y πr (n) = (−pr ), p|n

where p ranges over all prime divisors of n. We denote by δr the multiplicative function πr Jr , that is, for all n ∈ N, Y (1 − pr ). δr (n) = nr p|n

In particular, δ1 (n) = ϕ(n)π1 (n) = n

Q

(1 − p).

p|n

Let c be a fixed positive integer. We denote by Gc the arithmetic function defined by Gc (n) = gcd(c, n) = (c, n) for all n ∈ N. In particular, G1 = I. The function Gc is multiplicative (see Proposition 6.9 in Appendix). The arithmetic function n 7→ [c, n] is clearly not multiplicative. We denote by Hc the arithmetic function defined by n 1 Hc (n) = [c, n] = , c (c, n)

for n ∈ N.

This function is multiplicative (see Proposition 6.17). We consider also the next multiplicative function, denoted by εc , defined by ( 1, if n | c, εc (n) = 0, otherwise. In particular, ε1 = e. For example, ε12 (n) = 1 for n ∈ {1, 2, 3, 4, 6, 12} and ε12 (n) = 0 in other cases. In Appendix some properties of the inverse ε−1 c are given.

3

Gcd-determinants

Let c be a fixed positive integer. Let us recall that if n ∈ N, then Gc (n) = gcd(c, n) = (c, n), εc (n) = 1 when n | c, and εc (n) = 0 when n - c. Observe that if a, b are coprime positive integers, then the numbers Gc (a) and Gc (b) are coprime, too. Thisimplies that if f is a multiplicative function, then the function f ◦ Gc , that is, n 7→ f (c, n) , is also multiplicative. Proposition 3.1. Let f, g be multiplicative functions such that f = g ∗ I. Let c ∈ N, and let Fc = f ◦ Gc . Then Fc ∗ µ = εc g, that is, if n ∈ N, then ( g(n), if n | c, (Fc ∗ µ) (n) = 0, otherwise.



Proof. Put h := Fc ∗ µ, and let n ∈ N. Step 1. Assume that n | c. In this case h(n) = g(n). Indeed, P P n h(n) = (Fc ∗ µ) (n) = Fc (d)µ d = f (c, d) µ d|n

=

P

=

f (d)µ

d|n

n d

d|n

n d

5

= (f ∗ µ) (n) = (g ∗ I) ∗ µ (n)

g ∗ (I ∗ µ) (n) = g(n).

Thus, if n | c then h(n) = εc (n)g(n) = (εc g) (n). Step 2. Assume that n > 2 and (c, n) = 1. Then n - c. In this case h(n) = 0. Indeed, P P h(n) = (Fc ∗ µ) (n) = Fc (d)µ nd = f (c, d) µ nd d|n

d|n

=

P

f (1)µ

d|n

n d

=

P

µ(d) = 0.

d|n

We applied the equality µ ∗ I = e. Thus, if n > 2 and (c, n) = 1, then h(n) = 0 = 0 · g(n) = εc (n)g(n) = (εc g) (n). Step 3. Assume that n is a prime power pα with α > 1, and c < n. We shall show that h(n) = 0. Indeed, if p - c then (c, n) = 1 and the assertion follows from Step 1. Assume that p | c. Let c = pβ u, p - u, 1 6 β < α. Then P α h(n) = (Fc ∗ µ) (n) = Fc (d)µ pd = Fc (pα ) µ(1) + Fc (pα−1 ) µ(p) d|pα

= f (c, pα ) − f (c, pα−1 ) = f (pβ u, pα ) − f (pβ u, pα−1 ) = f pβ − f pβ = 0. Thus, if n = pα with α > 1, and c < n, then h(n) = 0 = 0·g(n) = εc (n)g(n) = (εc g) (n). Step 4. Assume that n > 2 and n | c. We shall show that h(n) = 0. For this aim consider the prime decomposition n = pα1 1 · · · pαs s . Since n - c, there exists an i ∈ {1, . . . , s} such that pαi i - c. Put p := pi and α := αi . Let c = upα + v, where u, v ∈ Z and 1 6 v < pα . Then h(pα ) = 0. Indeed, using Step 3, we have P P α α α h(pα ) = (Fc ∗ µ) (pα ) = Fc (d)µ pd = f (up + v, d) µ pd d|pα d|pα P P α α = f (v, d) µ pd = Fv (d)µ pd = (Fv ∗ µ) (pα ) = 0. d|pα

d|pα

Hence, h (pαi i ) = 0. But the function h is multiplicative, so h(n) = h (pα1 1 · · · pαs s ) = h (pα1 1 ) · · · h (pαi i ) · · · h (pαs s ) = 0. Thus, in any case we have the equality h(n) = (εc g) (n). This completes the proof. Proposition 3.2. Let f, g be multiplicative functions such that f = g ∗ I, and let S = {x1 , . . . , xn } be a factor-closed set of positive integers. Assume that x1 < x2 < · · · < xn . Then for every i ∈ {1, . . . , n} the following equalities hold.  x  g(xn ), if i = n, X n f (xi , xk ) µ =  0, xk otherwise. xk |xn



6

Proof. Let i ∈ {1, . . . , n}, and let Fxi = f ◦ Gxi , that is, Fxi (m) = f (xi , m) for all m ∈ N. Then we have X x X x X xn n n f (xi , xk ) µ = Fxi (xk )µ = Fxi (d)µ = (Fxi ∗ µ) (xn ). xk xk d xk |xn

xk |xn

d|xn

Thus, this assertion follows from Proposition 3.1.

3.1

Smith’s theorem

In 1875 J. H. S. Smith [32] proved the following theorem. Theorem 3.3 (Smith 1875). Let S = {x1 , . . . , xn } (where n > 1) be a factor-closed set of positive integers with x1 < x2 < · · · < xn . If f, g : N → C are multiplicative functions such that f = g ∗ I, then h i det f (xi , xj ) = g(x1 )g(x2 ) · · · g(xn ). 16i,j6n

In other words, if f, g : N → C are functions such that f (m) =

X

g(d) for m ∈ N,

d|m

then f (x , x ) f (x , x ) ··· 1 1 1 2 f (x2 , x1 ) f (x2 , x2 ) · · · .. .. . . f (xn , x1 ) f (xn , x2 ) · · ·

f (x1 , xn ) f (x2 , xn ) .. . f (xn , xn )

= g(x1 )g(x2 ) · · · g(xn ).

h i Proof. (Smith [32]). Denote the matrix f (xi , xj ) by Mn . Let Dn = det Mn , and let A1 , . . . , An be the columns of the matrix Mn . Replace the last column An by the sum X xn Ak . µ xk xk |xn

Then the value of Dn is not changed and, by Proposition 3.2, the last new column is equal to [0, 0, . . . , 0, g(xn )]T . Hence, Dn = g(xn )Dn−1 and consequently, Dn = g(xn )g(xn−1 ) · · · g(x2 )g(x1 ). If S = {1, 2, . . . , n}, then the above theorem has the following form. Theorem 3.4 (Smith 1875). Let n be a positive integer. If f, g : N → C are functions such that f = g ∗ I, then h i det f (i, j) = g(1)g(2) · · · g(n). 16i,j6n



In other words, if f, g : N → C are functions such that f (m) =

X

7

g(d) for m ∈ N,

d|m

then

f (1, 2) · · · f (1, 1) f (2, 1) f (2, 2) ··· .. .. . . f (n, 1) f (n, 2) · · ·

f (1, n) f (2, n) .. . f (n, n)

= g(1)g(2) · · · g(n).

Proofs of this form can be found in several references ([32, 9, 10, 29]). Of course it is an immediate consequence of Theorem 3.3. We present a second proof. Proof. ([9]). Denote this determinant by D. Consider the numbers aij defined by ( 1, when j | i, aij = 0, when j - i. n X X Then, for all i, j ∈ N, we have f i, j) = g(d) = aik ajk g(k), and this implies k=1

d|(i,j)

that D = A · B, where a11 a12 · · · a21 a22 · · · A = .. .. . . an1 an2 · · ·

a1n a2n .. . ann

,

a11 g(1) a21 g(1) · · · a12 g(2) a22 g(2) · · · B= .. .. . . a1n g(n) a2n g(n) · · ·

. ann g(n) an1 g(1) an2 g(2) .. .

Observe that aii = 1 and aij = 0 for i < j. Hence, A = 1 and B = g(1)g(2) · · · g(n). Therefore, D = g(1)g(2) · · · g(n). h i Proposition 3.5. Let Dn = det f (i, j)

16i,j6n

, where f : N → C is a function. If

f (p) = f (1) for a prime number p, then Dn = 0 for all n > p. Proof. Observe that f = g ∗ I, where g = f ∗ µ. Since f (p) = f (1), we have g(p) = 0 because g(p) = f (1)µ(p) + f (p)µ(1) = −f (1) + f (p) = 0. Hence, if n > p then, by Theorem 3.3, we have Dn = g(1) · · · g(p) · · · g(n) = 0. h i Example 3.6. Let Dn = det f (i, j)

, where f (x) = x2 − 4x + 20. Then 16i,j6n

D1 = 17, D2 = −17, and Dn = 0 for n > 3. It follows from Proposition 3.3, because f (1) = f (3). h i Example 3.7. Let Dn = det f (i, j) , where f (x) = x2 − 8x + 100. Then 16i,j6n

Dn = 0 for n > 7. It follows from Proposition 3.3, because f (1) = f (7).


3.2


8

Smith determinants

As a consequence of Theorem 3.3 and the equality T = ϕ∗I, we obtain the following theorem Theorem 3.8

(Smith 1875). (x1 , x1 ) (x1 , x2 ) · · · (x2 , x1 ) (x2 , x2 ) · · · .. .. . . (xn , x1 ) (xn , x2 ) · · ·

= ϕ(x1 )ϕ(x2 ) · · · ϕ(xn ). (xn , xn ) (x1 , xn ) (x2 , xn ) .. .

If S = {1, 2, . . . , n}, then we obtain the following well known theorem (see for example [32, 14, 4, 22]). Theorem 3.9 ([32]). (1, 1) (1, 2) · · · (2, 1) (2, 2) · · · .. .. . . (n, 1) (n, 2) · · ·

= ϕ(1)ϕ(2) · · · ϕ(n). (n, n) (1, n) (2, n) .. .

Examples: D1 = D2 = 1, D3 = 2, D4 = 4, D5 = 16, D6 = 32, D7 = 26 3, D8 = 28 3, D9 = 29 32 , D10 = 211 32 , D11 = 212 32 5, D12 = 214 32 5. The above determinant is called the Smith determinant. Many generalizations of Smith determinants have been presented in literature, see [1, 5, 18, 19, 23, 30]. Dickson ([15] 122-129) reports on several papers devoted to proofs and extensions of Smith’s determinant. Let S = {x1 , . . . , xn } be a finite ordered set of distinct positive integers. We do not assume that S is factor-closed. The gcd matrix defined on S is given by [(xi , xj )] and is denoted by (S). In 1989, Scott Beslin and Steve Ligh [4] gave the conjecture that if det(S) = ϕ(x1 )ϕ(x2 ) · · · ϕ(xn ) then S is factor-closed. In 1990, Zhongshan Li [22] proved that this conjecture is true. Theorem 3.10 (Li [22]). Let S = {x1 , . . . , xn } be an ordered set of distinct positive integers. Then det(S) > ϕ(x1 )ϕ(x2 ) · · · ϕ(xn ), and the equality det(S) = ϕ(x1 )ϕ(x2 ) · · · ϕ(xn ) holds if and only if the set S is factor-closed. The matrix (S) is always positive definite ([4, 22]). Moreover (see [22]), 1 det(S) 6 x1 x2 · · · xn − n! . 2 Note also:



Proposition 3.11. Let S={x1 , . . . , xn } If S is factor closed, then ϕ (x1 , x2 ) · · · ϕ (x1 , x1 ) ϕ (x2 , x1 ) ϕ (x , x ) ··· 2 2 .. .. . . ϕ (x , x ) ϕ (x , x ) ··· n 1 n 2

9

be an ordered set of distinct positive integers. ϕ (x1 , xn ) ϕ (x2 , xn ) .. . ϕ (xn , xn )

= h(x1 )h(x2 ) · · · h(xn ),

where h = ϕ ∗ µ. Proof. This is an immediate consequence of Theorem 3.3, because ϕ = h ∗ I. ϕ (1, 2) · · · ϕ (1, 1) ϕ (2, 1) ϕ (2, 2) · · · Example 3.12. . .. . . . ϕ (n, 1) ϕ (n, 2) · · ·

ϕ (1, n) ϕ (2, n) .. . ϕ (n, n)

= 0 for n > 2.

Proof. It follows from Proposition 3.11 that this determinant equals h(1) · · · h(n), where h = ϕ∗µ. But h(2) = ϕ(1)µ(2)+ϕ(2)µ(1) = −1+1 = 0. Hence, the determinant equals 0 for n > 2. Proposition 3.13. Let f : N → C be the multiplicative function defined by f (pm ) = 1 + for each prime power gers. Then f (x1 , x1 ) f (x2 , x1 ) .. . f (x , x ) n 1

pm − 1 pm−1 (p − 1)2

pm . Let S = {x1 , . . . , xn } be a factor-closed set of positive inte f (x1 , x2 ) · · · f (x2 , x2 ) · · · .. . f (xn , x2 ) · · ·

f (x1 , xn ) f (x2 , xn ) .. . f (xn , xn )

1 = ϕ(x1 )ϕ(x2 ) · · · ϕ(xn ) .

Proof. Denote by g the function µ ∗ f . Then, for each prime power pm , we have g (pm ) = µ(1)f (pm ) + µ(p)f pm−1 = f (pm ) − f pm−1 =

1 pm−1 (p

− 1)

=

1 . ϕ(pm )

1 Hence, g(m) = ϕ(m) for all positive integer m, because the functions g and ϕ are multiplicative. Note that f = g ∗ I. Therefore, by Theorem 3.3, the determinant is 1 . equal to g(x1 ) · · · g(xn ) = ϕ(x1 )···ϕ(x n)


3.3


10

Power gcd-determinants

In this subsection r is a real number. Let us recall (see Section 2) that Jordan’s totient function is defined by Y 1 r 1− r Jr (n) = n p p|n

for n ∈ N, where p ranges over all prime divisors of n. In particular, J1 = ϕ and J0 = e. We know that Jr ∗ I = T r , where T r (n) = nr for n ∈ N (see Proposition 6.7). This implies that Jr = T r ∗ µ, Jr−1 = µT r ∗ I, where Jr−1 is the inverse of Jr with respect to Dirichlet convolution. Theorem 3.14 ([32, 11]). Let S={x1 , . . . , xn } be an ordered set of distinct positive integers, and let r be a real number. If S is factor closed, then (x1 , x1 )r (x1 , x2 )r · · · (x1 , xn )r (x2 , x1 )r (x2 , x2 )r · · · (x2 , xn )r = Jr (x1 )Jr (x2 ) · · · Jr (xn ). .. .. .. . . . (xn , x1 )r (xn , x2 )r · · · (xn , xn )r Proof. Use Theorem 3.3 and the equality Jr ∗ I = T r . For S = {1, 2, . . . , n} the above theorem has the following form. Theorem 3.15 ([9, 11]). If r is a real number and n is a positive integer, then (1, 1)r (1, 2)r · · · (1, n)r [n/p] (2, 1)r (2, 2)r · · · (2, n)r Y 1 r 1− r , = Jr (1)Jr (2) · · · Jr (n) = (n!) .. .. .. p . . . p (n, 1)r (n, 2)r · · · (n, n)r where p ranges over all primes belonging to the set {1, 2, . . . , n}. Note some special cases of the above theorems. We usually denote by Dn every determinant, which appears in the presented theorems and propositions. 3.16. Let Dn = det [(i, j)2 ]n×n = det [(i2 , j 2 )]n×n . Then [n/p] Y 1 2 Dn = J2 (1)J2 (2) · · · J2 (n) = (n!) 1− 2 , p p where p ranges over all primes belonging to {1, 2, . . . , n}. Examples: D1 = 1, D2 = 3, D3 = 24, D4 = 288 = 25 32 , D5 = 28 33 , D6 = 211 34 , D7 = 215 35 , D8 = 219 36 , D11 = 228 311 5. 3.17. Let S={x1 , . . . , xn } be an ordered set of distinct positive integers, and let r be a real number. If S is factor closed, then p p p (x1 , x1 ) (x , x ) · · · (x , x ) 1 2 1 n p (x , x ) p(x , x ) · · · p(x , x ) 2 1 2 2 2 n = J1/2 (x1 )J1/2 (x2 ) · · · J1/2 (xn ). .. .. .. . . . p p p (xn , x1 ) (xn , x2 ) · · · (xn , xn )


3.18. Let Dn = det


hp i (i, j) n×n

. Then Dn =

p Q (n!) 1−

√1 p

p

[n/p]

11

, where p ranges

over all primes belonging to {1, 2, . . . , n}. Proposition 3.19. Let S={x1 , . . . , xn } be an ordered set of distinct positive integers, and let r be a real number. If S is factor closed, then 1 1 1 · · · (x1 ,x1 ) (x1 ,x2 ) (x1 ,xn ) 1 (x ,x ) (x 1,x ) · · · (x 1,x ) 2 2 2 n 2 1 = J−1 (x1 )J−1 (x2 ) · · · J−1 (xn ). .. .. .. . . . 1 (xn ,x ) (xn1,x ) · · · (xn1,xn ) 1

3.20. Let Dn = det

2

h

1 (i,j)

i n×n

. Then Dn =

1 Y (1 − p)[n/p] . n! p

1 Examples: D1 = 1, D2 = − 21 , D3 = 31 , D4 = − 12 , D5 = 1 1 1 D8 = 420 , D9 = − 1890 , D10 = − 4725 . ([34], see Theorem 3.15) .

1 , 15

D6 =

1 , 45

2 D7 = − 105 ,

[n/p] 1 Y 1 − p2 . 2 n×n n×n (n!) p 3 2 96 Examples: D1 = 1, D2 = − 34 , D3 = 32 , D4 = − 18 , D5 = 25 , D6 = 25 , D7 = − 1225 , 4 8 9 D8 = 2450 , D9 = − 11025 , D10 = − 30625 . (see Theorem 3.15) . 3.21. Let Dn = det

h

1 (i,j)2

i

= det

h

1 (i2 ,j 2 )

i

. Then Dn =

3.22. If r is a real number, and {x1 , . . . , xn } is a factor-closed set of positive integers, then Jr (x1 , x2 ) · · · Jr (x1 , xn ) Jr (x1 , x1 ) Jr (x2 , x1 ) Jr (x2 , x2 ) · · · Jr (x2 , xn ) = h(x1 )h(x2 ) · · · h(xn ), .. .. .. . . . Jr (xn , x1 ) Jr (xn , x2 ) · · · Jr (xn ,n ) where h = Jr ∗ µ is the multiplicative function defined, for prime powers pm , by h(p) = 2 pr − 2 and h(pm ) = pmr 1 − p1r for m > 2.

3.4

Gcd-determinants with tau

Let us recall that τ (n) is the number of all positive divisors of n. Proposition 3.23. If {x1 , . . . , xn } is a factor-closed set of positive integers, then τ (x1 , x2 ) · · · τ (x1 , xn ) τ (x1 , x1 ) τ (x2 , x1 ) τ (x2 , x2 ) · · · τ (x2 , xn ) = 1. .. .. .. . . . τ (xn , x1 ) τ (xn , x2 ) · · · τ (xn , xn )



12

Proof. It follows from Theorem 3.3, because τ = I ∗ I. Proposition 3.24 ([31]). The determinant τ (2, 2) τ (2, 3) ··· τ (3, 2) τ (3, 3) · · · . .. . . . τ (n, 2) τ (n, 3) · · ·

τ (2, n) τ (3, n) .. . τ (n, n)

is equal to the number of all square-free numbers belonging to {1, 2, . . . , n}. Proof. Let dij = τ (i, j) for all i, j ∈ {1, . . . , n}. Consider the matrices     1 d12 d13 · · · d1n d22 d23 · · · d2n  0 d22 d23 · · · d2n   d32 d33 · · · d3n      0 d32 d33 · · · d3n   D =  .. , E =  . .. ..    . .. .. ..  . .   . . .  dn2 dn3 · · · dnn 0 dn2 dn3 · · · dnn Since det D = det E, we need to show that det E is equal to the number of all squarefree numbers belonging to {1, 2, . . . , n}. Denote by Ei the i-th row of E. Beginning with i = n and proceeding towards i = 2, replace each Ei , by X i µ Ek . k k|i P Let γij = 1 if i | j, and γij = 0 if i - j. Observe that γkj = dij . Hence, using the k|i P Möbius inversion formula we obtain µ ki dkj = γij . This implies that det E = det F , k|i

where

    F =  

1 γ12 γ13 µ(2) γ22 γ23 µ(3) γ32 γ33 .. .. . . µ(n) γn2 γn3

··· ··· ···

γ1n γ2n γ3n .. .

···

γnn

    .  

Denote by Fi the i-th row of F . Let G = [gij ]16i,j6n be the matrix obtained from F n P by replacing F1 by µ(i)Fi . Then det G = det F = det D. If j > 1, then g1j = 0. In i=1

fact, g1j =

n X i=1

Moreover, g11 =

n P

µ(i)γij =

X

µ(i) = 0.

i|j

µ(i)2 is equal to the number of all square-free integers belonging

i=1

to {1, 2, . . . , n}. Observe that [gij ]26i,j6n is an upper triangular matrix with 1’s on the diagonal. Hence, det D = det G = g11 . Thus det D is equal to the number of all square-free integers belonging to {1, 2, . . . , n}. The next proposition is an immediate consequence of Theorem 3.3.



Proposition 3.25. Let {x1 , . . . , xn } be If h = I ∗ I ∗ I = τ ∗ I, then h (x , x ) h (x , x ) ··· 1 1 1 2 h (x2 , x1 ) h (x2 , x2 ) · · · . .. . . . h (xn , x1 ) h (xn , x2 ) · · ·

3.5

13

a factor-closed set of positive integers. h (x1 , xn ) h (x2 , xn ) .. . h (xn , xn )

= τ (x1 )τ (x2 ) · · · τ (xn ).

Gcd-determinants with sigma

Let us recall that σ(n) is the sum of all positive divisors of n. If r is a real number, then σr (n) is the sum of the powers r of all positive divisors of n. In particular, σ1 = σ. Note that σ = T ∗ I and σr = T r ∗ I. Hence, immediately from Theorem 3.3 we obtain the following two propositions. Proposition 3.26. If r positive integers, then σr (x1 , x1 ) σr (x2 , x1 ) .. . σr (xn , x1 )

is a real number and {x1 , . . . , xn } is a factor-closed set of σr (x1 , x2 ) · · · σr (x2 , x2 ) · · · .. . σr (xn , x2 ) · · ·

Proposition 3.27. If {x1 , . . . , xn } is a σ (x1 , x2 ) σ (x1 , x1 ) σ (x2 , x1 ) σ (x , x ) 2 2 . . . . . . σ (xn , x1 ) σ (xn , x2 )

σr (x1 , xn ) σr (x2 , xn ) .. . σr (xn , xn )

= (x1 x2 . . . xn )r .

factor-closed set of positive integers, then · · · σ (x1 , xn ) · · · σ (x2 , xn ) = x 1 x 2 . . . xn . .. . · · · σ (xn , xn )

Note some special cases of the above propositions, which appear in [32, 14, 9]. σ (1, 2) · · · σ (1, n) σ (1, 1) σ (2, 1) σ (2, 2) · · · σ (2, n) 3.28. = n!. .. .. .. . . . σ (n, 1) σ (n, 2) · · · σ (n, n) σr (1, 2) · · · σr (1, 1) σr (2, 1) σr (2, 2) · · · 3.29. . .. . . . σr (n, 1) σr (n, 2) · · ·

σr (1, n) σr (2, n) r = (n!) . .. . σr (n, n)



σ−1 (1, 2) · · · σ−1 (1, 1) σ−1 (2, 1) σ−1 (2, 2) · · · 3.30. .. .. . . σ−1 (n, 1) σ−1 (n, 2) · · ·

σ−1 (1, n) σ−1 (2, n) .. . σ−1 (n, n)

14

= 1. n!

3.31. If f : N → C is the multiplicative function defined by f (pm ) = prime power pm , then f (1, 1) f (1, 2) · · · f (1, n) f (2, 1) f (2, 2) · · · f (2, n) 1 = n! . .. .. .. . . . f (n, 1) f (n, 2) · · · f (n, n)

pm+1 −1 pm (p−1)

for each

Proof. Denote by g the function µ ∗ f . Then, for each prime power pm , we have pm+1 − 1 pm − 1 1 g (pm ) = µ(1)f (pm ) + µ(p)f pm−1 = m − m−1 = m. p (p − 1) p (p − 1) p Hence, g(m) = m1 for all positive integer m, because g multiplicative. Note that f = g ∗ I. Therefore, by Theorem 3.3, the determinant is equal to g(1) · · · g(n) = n!1 . The next propositions are immediate consequences of Theorem 3.3. Proposition 3.32. If r is a real positive integers, then h (x1 , x2 ) h (x1 , x1 ) h (x2 , x1 ) h (x2 , x2 ) . .. . . . h (xn , x1 ) h (xn , x2 )

number and {x1 , . . . , xn } is a factor-closed set of

··· ··· ···

h (x1 , xn ) h (x2 , xn ) .. . h (xn , xn )

= σr (x1 )σr (x2 ) . . . σr (xn ),

where h = σr ∗ I. Proposition

3.33. If h = σ ∗ I, then h (1, 1) h (1, 2) · · · h (2, 1) h (2, 2) · · · .. .. . . h (n, 1) h (n, 2) · · ·

h (1, n) h (2, n) .. . h (n, n)

= σ(1)σ(2) · · · σ(n).


3.6


15

Other gcd-determinants

Recall that e(1) = 1 and e(n) = 0 for n > 2, and moreover, e = µ ∗ I. Hence, by Theorem 3.3, we obtain Proposition 3.34. If e (x1 , x1 ) e (x2 , x1 ) .. . e (xn , x1 ) Proposition

{x1 , . . . , xn } e (x1 , x2 ) e (x2 , x2 ) .. . e (xn , x2 )

is a factor-closed · · · e (x1 , xn ) · · · e (x2 , xn ) .. . · · · e (xn , xn )

3.35. e (1, 1) e (1, 2) · · · e (2, 1) e (2, 2) · · · .. .. . . e (n, 1) e (n, 2) · · ·

e (1, n) e (2, n) .. . e (n, n)

set of positive integers, then = µ(x1 )µ(x2 ) . . . µ(xn ).

= µ(1)µ(2) · · · µ(n).

In particular: D1 = 1, D2 = −1, D3 = 1, and Dn = 0 for n > 4. Proposition 3.36. If µ (x1 , x1 ) µ (x2 , x1 ) .. . µ (xn , x1 )

{x1 , . . . , xn } µ (x1 , x2 ) µ (x2 , x2 ) .. . µ (xn , x2 )

is a factor-closed set of positive integers, then · · · µ (x1 , xn ) · · · µ (x2 , xn ) = h(x1 )h(x2 ) . . . h(xn ), .. . · · · µ (xn , xn )

where h = µ ∗ µ = τ −1 . Colorary 3.37. Let µ (1, 2) · · · µ (1, 1) µ (2, 1) µ (2, 2) ··· Dn = .. .. . . µ (n, 1) µ (n, 2) · · ·

µ (1, n) µ (2, n) .. . µ (n, n)

.

Then D1 = 1, D2 = −2, D3 = 22 , D4 = 22 , D5 = −23 , D6 = −25 , and D7 = 26 , and Dn = 0 for n > 8. In the next gcd-determinants appears the multiplicative function πr . The following propositions are consequences of Theorem 3.3.



16

Proposition 3.38. If r is a real number, and {x1 , . . . , xn } is a factor-closed set of positive integers, then πr (x1 , x2 ) · · · πr (x1 , xn ) πr (x1 , x1 ) πr (x2 , x1 ) πr (x2 , x2 ) · · · πr (x2 , xn ) = h(x1 )h(x2 ) . . . h(xn ), .. .. .. . . . πr (xn , x1 ) πr (xn , x2 ) · · · πr (xn , xn ) where h is the multiplicative function defined, for each prime power pm , by h(pm ) = −(1 + pr ) for m = 1, and h(pm ) = 0 for m > 1. Colorary 3.39. πr (1, 2) · · · πr (1, 1) πr (2, 1) π (2, 2) ··· r .. .. . . πr (n, 1) πr (n, 2) · · ·

πr (1, n) πr (2, n) .. . πr (n, n)

= 0 for n > 4.

Proposition 3.40. Let r be a real number and let {x1 , . . . , xn } be a factor-closed set of positive integers. Let f = πr ∗ I. Then f is the multiplicative function defined by f (pm ) = 1 − mpr , for each prime power pm , and we have f (x1 , x2 ) · · · f (x1 , xn ) f (x1 , x1 ) f (x2 , x1 ) f (x2 , x2 ) · · · f (x2 , xn ) = πr (x1 )πr (x2 ) · · · πr (xn ). .. .. .. . . . f (xn , x1 ) f (xn , x2 ) · · · f (xn , xn ) Colorary 3.41.et r be a real number and let f = πr ∗ I. Then f (1, 2) · · · f (1, n) f (1, 1) Y f (2, 1) f (2, 2) · · · f (2, n) (−pr )[n/p] . = π (1)π (2) · · · π (n) = r r r . . . . . . p . . . f (n, 1) f (n, 2) · · · f (n, n)

4

Lcm-determinants

Let us recall (see Section 2) that if r is a real number, then πr is the multiplicative function defined by πr (pm ) = −pr , for each prime power pm . We denote by δr the multiplicative function Jr πr , that is, δr (n) = Jr (n)πr (n) for n ∈ N. In particular, Y Y δ1 (n) = ϕ(n)π1 (n) = n (1 − p), δr (n) = nr (1 − pr ). p|n

p|n


4.1


17

Smith lcm-determinants

Let us start with the following well known theorem (see, for example [32, 3, 7, 27, 34]). Theorem 4.1 (Smith 1875). [1, 1] [1, 2] [1, 3] · · · [2, 1] [2, 2] [2, 3] · · · [3, 1] [3, 2] [3, 3] · · · .. .. .. . . . [n, 1] [n, 2] [n, 3] · · ·

Y = δ1 (1)δ1 (2) · · · δ1 (n) = n! (1 − p)[n/p] , p [n, n] [1, n] [2, n] [3, n] .. .

where p ranges over all primes belonging to the set {1, 2, . . . , n}. Examples: D1 = 1, D2 = −2, D3 = 12, D4 = −48, D5 = 960 = 26 · 3 · 5, D6 = 11520 = 28 · 32 · 5. h i Proof. ([34]). Let A, B, N be the n × n matrices defined by A = [i, j] , 16i,j6n i h i h 1 and N = nij , where nii = i and nij = 0 for i 6= j. Then B = (i,j) 16i,j6n 16i,j6nQ A = N BN , det N = n!, and det B = n!1 (1 − p)[n/p] (see 3.20). Hence, p

det A = (n!)2 det B = n!

Y

(1 − p)[n/p] .

p

The equality

n Q k=1

δ1 (k) = n!

Q

(1 − p)[n/p] is obvious.

p

The above determinant is called the Smith lcm-determinant. Several generalizations of this determinants have been presented in literature, see for example [11, 20]. Smith [32] observed that this result remains valid if the set {1, 2, . . . , n} is replaced by a factor-closed set (see also [7, 8, 20]). Theorem 4.2 ([32]). Let S = {x1 , . . . , xn } be integers. If S is factor-closed, then [x1 , x1 ] [x1 , x2 ] [x1 , x3 ] · · · [x1 , xn ] [x2 , x1 ] [x2 , x2 ] [x2 , x3 ] · · · [x2 , xn ] .. .. .. .. . . . . [xn , x1 ] [xn , x2 ] [xn , x3 ] · · · [xn , xn ]

an ordered set of distinct positive = δ1 (x1 )δ2 (x2 ) · · · δ1 (xn ).

Proof. We do a small modification of the proof of Theorem 4.1. We use the xx equalities [xi , xj ] = (xii,xjj ) and we apply results from the previous section. It follows from this theorem that if S is factor-closed then the above determinant is nonzero. In a general case, when S is not factor-closed, this determinant may not be nonzero (see [7]). For example, if S = {1, 2, 15, 42}, then 1 2 15 42 2 2 30 42 = 0. det[S] = 15 30 15 210 42 42 210 42 The same we have for S = {1, 2, 3, 4, 5, 6, 10, 45, 180}. This set is gcd-closed but not factor-closed ([19]).


4.2


18

Power lcm-determinants

Theorem 4.3 ([32]). If [1, 1]r [1, 2]r · · · [2, 1]r [2, 2]r · · · [3, 1]r [3, 2]r · · · .. .. . . [n, 1]r [n, 2]r · · ·

r is a real number, then [1, n]r [2, n]r Y [3, n]r = δr (1)δr (2) · · · δr (n) = (n!)r (1 − pr )[n/p] , .. p . r [n, n]

where δr = πr Jr , and where p ranges over all primes belonging to the set {1, 2, . . . , n}. h i r Proof. Let A, B, N be the n × n matrices defined by A = [i, j] , B = i h i h (i, j)−r , N = nij , where N is the diagonal matrix with nii = ir and nij = 0 for Q i 6= j. Then A = N BN , det N = (n!)r , and B = (n!)−r (1 − pr )[n/p] (see Theorem p Q 3.15). Hence, det A = (n!)2r det B = (n!)r (1 − pr )[n/p] . p

Colorary 4.4. 1 1 [1,1] [1,2] · · · 1 1 [2,1] [2,2] ··· .. .. . . 1 1 [n,1] [n,2] · · ·

1 [1,n] 1 [2,n]

.. .

1 [n,n]

[n/p] Y 1 1 = δ−1 (1)δ−1 · · · δ−1 (n) = 1− , n! p p

where δ−1 = π−1 J−1 , and where p ranges over all primes belonging to the set {1, 2, . . . , n}. 1 1 1 1 1 , D4 = 144 , D5 = 900 , D6 = 16200 , D7 = 132300 , Examples: D1 = 1, D2 = 14 , D3 = 18 1 D8 = 2116800 . Observe that Dn > 0 for every n. Note also Theorem 4.5 ([32, 11]). Let S={x1 , . . . , xn } be an ordered set of distinct positive integers, and let r be a real number. If S is factor closed, then [x1 , x1 ]r [x1 , x2 ]r · · · [x1 , xn ]r [x2 , x1 ]r [x2 , x2 ]r · · · [x2 , xn ]r [x3 , x1 ]r [x3 , x2 ]r · · · [x3 , xn ]r = δr (x1 )δr (x2 ) · · · δr (xn ). .. .. .. . . . r r r [xn , x1 ] [xn , x2 ] · · · [xn , xn ] Proof. We do a small modification of the proof of Theorem 4.3. We use the xx equalities [xi , xj ] = (xii,xjj ) and we apply results from the previous section.


4.3


19

Examples of lcm-determinants

We start with the following simple lemma. Lemma 4.6. If n ∈ N, then ϕ [1, n] = ϕ [2, n] . Proof. Of course [1, n] = n. [2, n]=n if 2 | n, and [2, n] = 2n Moreover, if 2 - n. Hence, if 2 | n, then ϕ [2, n] = ϕ(n) = ϕ [1, n] , and if 2 - n, then ϕ [2, n] = ϕ(2n) = ϕ(2)ϕ(n) = 1 · ϕ(n) = ϕ [1, n] . ϕ [1, 1] ϕ [1, 2] · · · ϕ [2, 1] ϕ [2, 2] · · · Example 4.7. .. .. . . ϕ [n, 1] ϕ [n, 2] · · ·

ϕ [1, n] ϕ [2, n] .. . ϕ [n, n]

= 0 for n > 2.

Proof. Let n > 2. By Lemma 4.6, the first row is equal to the second row. Hence, the determinant equals zero. Example 4.8.

1 ϕ([1,1]) 1 ϕ([2,1])

1 ϕ([1,2]) 1 ϕ([2,2])

···

1 ϕ([n,1])

1 ϕ([n,2])

···

.. .

···

.. .

1 ϕ([1,n]) 1 ϕ([2,n])

.. .

1 ϕ([n,n])

= 0 for n > 2.

Proof. Let n > 2. By Lemma 4.6, the first row is equal to the second row. Hence, the determinant equals zero.

4.4

Initial values of some lcm-determinants

We denote by Dn every τ [1, 1] τ [2, 1] 4.9. Dn = .. . τ [n, 1]

determinant, which appears in τ [1, 2] · · · τ [1, n] τ [2, 2] · · · τ [2, n] , .. .. . . τ [n, 2] · · · τ [n, n]

σ [1, 1] σ [1, 2] · · · σ [2, 1] σ [2, 2] · · · 4.10. Dn = .. .. . . σ [n, 1] σ [n, 2] · · ·

σ [1, n] σ [2, n] , .. . σ [n, n]

the presented examples. D1 D2 D3 D4 D5

= 1, = −2, = 4, = −6, = 12,

D1 D2 D3 D4 D5 D6

D6 = 48, D7 = −96, D8 = 128, D9 = −192, D10 = −768.

= 1, = −6, = 72, = −672 = −25 · 3 · 7, = 20160 = 26 · 32 · 5 · 7, = 1451520 = 29 · 34 · 5 · 7.



µ [1, 1] µ [1, 2] · · · µ [2, 1] µ [2, 2] · · · 4.11. Dn = .. .. . . µ [n, 1] µ [n, 2] · · ·

µ [1, n] µ [2, n] , .. . µ [n, n]

D1 D2 D3 D4 D5 D6

= 1, = −2, = 4, = 0, = 0, = 0.

1/τ ([1, 1]) 1/τ ([1, 2]) · · · 1/τ ([2, 1]) 1/τ ([2, 2]) · · · 4.12. Dn = .. .. . . 1/τ ([n, 1]) 1/τ ([n, 2]) · · ·

1/τ ([1, n]) 1/τ ([2, n]) .. .

, 1/τ ([n, n])

D1 D2 D3 D4 D5

= 1, = 1/4, = 1/16, = 1/144, = 1/576.

1/σ([1, 1]) 1/σ([1, 2]) · · · 1/σ([2, 1]) 1/σ([2, 2]) · · · 4.13. Dn = .. .. . . 1/σ([n, 1]) 1/σ([n, 2]) · · ·

, 1/σ([n, n])

D1 D2 D3 D4 D5

= 1, = 2/9, = 1/24, = 1/294, = 5/254016.

5

20

1/σ([1, n]) 1/σ([2, n]) .. .

Determinants with some arithmetic functions

We present initial values of some determinants Dn . ϕ(1) D1 ϕ(2) · · · ϕ(n) ϕ(2) ϕ(3) · · · ϕ(n + 1) D2 5.1. Dn = .. ; D3 .. .. . . . D4 ϕ(n) ϕ(n + 1) · · · ϕ(2n − 1) D 5

= 1, = 1, = 0, = −4, = 64,

τ (n) τ (n + 1) .. .

D1 D2 D3 D4 D5

= 1, = −2, = 3, = 3, = −7,

σ(1) σ(2) ··· σ(2) σ(3) ··· 5.3. Dn = .. .. . . σ(n) σ(n + 1) · · ·

σ(n) σ(n + 1) .. .

; σ(2n − 1)

D1 D2 D3 D4 D5

= 1, = −5, = 25, = 105, = −784,

µ(1) µ(2) ··· µ(2) µ(3) ··· 5.4. Dn = .. .. . . µ(n) µ(n + 1) · · ·

; µ(2n − 1)

D1 D2 D3 D4 D5

= 1, = −2, = 3, = 3, = −7,

τ (1) τ (2) ··· τ (2) τ (3) ··· 5.2. Dn = .. .. . . τ (n) τ (n + 1) · · ·

; τ (2n − 1)

µ(n) µ(n + 1) .. .

D6 = −24 · 3 · 5, D7 = −25 · 83, D8 = 210 · 3 · 5, D9 = −211 · 43, D10 = −211 · 33 · 31. D6 = −3, D7 = 16, D8 = 3, D9 = −27, D10 = −27. D6 = −22 · 5 · 7 · 31, D − 7 = 3 · 52 · 409, D8 = 2 · 3 · 53 · 229.

D6 = −5, D7 = 12, D8 = −19, D9 = −52, D10 = −52.


6


21

Appendix

6.1

Arithmetic functions

Let us recall (see Section 2) that if f, g : N → C are arithmetic functions, then we denote by f ∗ g the Dirichlet convolution of f and g. If p is a prime number, then (f ∗ g)(p) = f (1)g(p) + f (p)g(1), and more general, for all r ∈ N we have r

(f ∗ g)(p ) =

r X

f (pk )g(pr−k ).

k=0

Let us recall also that if a function f ∈ A is invertible in A, then we denote by f −1 the inverse of f with respect to the Dirichlet convolution. In particular, if f (1) = 1, then f −1 (1) = 1, and f −1 (p) = −f (p), when p is a prime, and more general, for all n ∈ N, we have n−1 X −1 n f (p ) = − f −1 (pk )f (pn−k ). k=0

Proposition 6.1 ([24] 43). For all f ∈ A and n ∈ N, we have the equality n n h i X X n f (i). (f ∗ I)(i) = i i=1 i=1

Proof. We use the following obvious equalities for all n, k ∈ N: ( hni n − 1 1, if k | n, − = k k 0, if k - n. Let F (n) :=

n h i X n i=1

i

f (i) and G(n) := (f ∗ I)(n) =

X

f (k).

k|n

Observe that G(n) = F (n) − F (n − 1). In fact, F (n) − F (n − 1) =

=

n h i X n

i

i=1 n h X i=1

Thus

n X

(f ∗ I)(i) =

i=1

F (n) =

n h i X n i=1

i

n X

f (i) −

n−1 X n−1 i=1

i

f (i) =

n h i X n i=1

i

f (i) −

n X n−1 i=1

i

f (i)

X ni n−1 − f (i) = f (k) = G(n). i i k|n

G(i) = F (1) + F (2) − F (1) + · · · + F (n) − F (n − 1) =

i=1

f (i). This completes the proof.

Recall that a multiplicative function is an arithmetic function f with the property f (1) = 1 and whenever a and b are coprime, then f (ab) = f (a)f (b). Observe that if a function f is multiplicative and p is a prime, then f −1 (p) = −f (p) and f −1 (p2 ) = f (p)2 − f (p2 ).

Andrzej Nowicki, 2017, Proposition 6.2.


([26] 308).

22

If f is a multiplicative function then, for all a, b ∈ N, f [a, b] f (a, b) = f (a)f (b).

Proof. Let a, b ∈ N. Let a = pα1 1 · · · pαnn and b = pβ1 1 · · · pβnn , where p1 , . . . , pn are distinct primes and α1 , . . . , αn , β1 , . . . , βn are nonnegative integers. Then (a,b) = γ1 δ p1 · · · pγnn and [a, b] = p11 · · · pδnn , where γi = min αi , βi and δi = max αi , βi for i = 1, . . . , n. Hence, enough if p is a prime and s, t are nonnegative itis to check that s t min(s,t) max(s,t) integers, then f p f p = f p f p . But it is obvious. Proposition 6.3. ([21] 464). Let f be a multiplicative function, and let d ∈ N with f (d) 6= 0. Then the function g defined by n 7→ ff(dn) , is multiplicative. (d) Proof. then

Let a, b ∈ N, (a, b) = 1. Then dab = d[a, b] = [da, db], (da, db) = d and

g(ab) =

f ([da, db]) f (ad)f (bd) f (da f (db) f (dab) = = = · = g(a)g(b). f (b) f (d) f ((a, b))f (d) f (d) f (d)

We used Proposition 6.2. Proposition 6.4. If f : N → C is a multiplicative function, then Y µf ∗ I (m) = 1 − f (p) p|m

for all positive integers m, where p ranges over all prime divisors of m. Proof. It is obvious for m = 1. Now let m > 2, and let m = pα1 1 · · · pαr r be the prime decomposition of m. Denote by h the function µf . Then we have:

α1 P α2 αr P P P h(d) = µf ∗ I (m) = h ∗ I (m) = ··· h pi11 · · · pirr i1 =0 i2 =0

d|m

=

α1 P α2 P

···

i1 =0 i2 =0 α1 P

=

i1 =0

=

αr P ir =0

α1 P

!

ir =0

h pi11 · · · h pirr =

! µf (pi11 ) · · ·

µf (pirr )

α1 P i1 =0

! h(pi11 ) · · ·

ir =0

p|m

This completes the proof. Note the following consequence of the above proposition Proposition 6.5. If f : N → C is a multiplicative function, then [ np ] Y µf ∗ I (m) = 1 − f (p) ,

m=1

where p ranges over all primes.

p

! h(pirr )

ir =0

= 1 + µ(p1 )f (p1 ) · · · 1 + µ(pr )f (pr )

Q 1 − f (p1 ) · · · 1 − f (pr ) = 1 − f (p) .

n Y

α1 P



23

Let us recall (see Section 2) that an arithmetic function f is said to be completely multiplicative if f (1) = 1 and f (ab) = f (a)f (b) holds for all positive integers a and b, even when they are not coprime. Proposition 6.6. If f is a completely multiplicative function, then f has the inverse f −1 , with respect to Dirichlet convolution, and then f −1 = µf . Proof. We need to show that µf ∗ f = e. Note that the functions µf ∗ f and e are multiplicative. Thus, it is enough to show that (µf ∗ f ) (pm ) = 0 for all m ∈ N and all primes p. We have (µf ∗ f ) (pm ) =

P

(µf )(d)f (pm /d) =

d|pm

m P

(µf )(pk )f (pm−k )

k=0

= µ(1)f (1)f (pm ) + µ(p)f (p)f (pm−1 ) = f (pm ) − f (pm ) = 0. This completes the proof. The next proposition is on Jordan’s totient function Jr . Proposition 6.7. Jr ∗ I = T r . Proof. The functions Jr ∗ I and T r are multiplicative. Thus, it is enough to show that (Jr ∗ I) (pm ) = pmr for all m ∈ N and all primes p. Let us verify: X m m m X X 1 kr k m p Jr (p ) = 1− r = (Jr ∗ I) (p ) = pkr − p(k−1)r = pmr . p k=0 k=0 k=0 This completes the proof. Let us recall that if r is a real number, then we denote by πr the multiplicative function defined by Y πr (n) = (−pr ), p|n

where p ranges over all prime divisors of n. It is easy to check that the inverse of πr with respect to Dirichlet convolution is the multiplicative function πr−1 : N → C defined by πr−1 (pm ) = pr (pr + 1)m−1 , for each prime power pm . Moreover, it is easy to prove the following proposition. Proposition 6.8. Let h = πr ∗ µ. Then πr = h ∗ I, and h is the multiplicative function defined, for each prime power pm , by ( −(1 + pr ), for m = 1, h (pm ) = 0, for m > 1. In other words, h is the function defined, for each positive integer n, by  1, when n = 1;    k Q h(n) = (−1)k (1 + pri ), when n = p1 . . . , pk , where p1 , . . . pk are distinct primes,  i=1   0, in other cases.



24

Let c be a fixed positive integer, and let Gc be the arithmetic function defined by Gc (n) = gcd(c, n) = (c, n) for all n ∈ N. In particular, G1 = I. Proposition 6.9. The function Gc is multiplicative.

h i Proof. It is well known that if u, v, w ∈ N, then [u, v], w = (u, w), (v, w) and moreover, if (u, v) = 1 then [u, v] = uv. Assume that a, b ∈ N with (a, b) = 1. Then the numbers (u, c), (v, c) are coprime, and we have: h i ab, c = [a, b], c = (a, c), (b, c) = (a, c)(b, c). Hence, Gc (ab) = Gc (a)Gc (b), when (a, b) = 1. If p is a prime number, then ( Gp (n) =

1, if p - n, p, if p | n.

−1 Consider the multiplicative function G−1 p , that is, Gp is the inverse of Gp with respect to Dirichlet convolution. It is easy to check that s s s−1 6.10. G−1 , for s > 1. p (p ) = (−1) p(p − 1)

6.11. If p, q are distinct primes and n ∈ N, then   1, for n = 0, −1 n −1, for n = 1, Gp (q ) =  0, for n > 2. 6.12.   1,     (−1)r ,       −1 (−1)s p(p − 1)s−1 , Gp (n) =    (−1)r+s p(p − 1)s−1 ,         0,

for n = 1, when p - n i n = p1 · · · pr , where p1 , . . . , pr are distinct primes, when n = ps , s > 1, when p | n i n = ps p1 · · · pr , s > 1, where p1 , . . . , pr are distinct primes different than p, in other cases.

s 6.13. Let n ∈ N. If p - n, then G−1 p (n) = µ(n). If p | n and n = p m with p - m, then s s−1 G−1 µ(m). p (n) = (−1) p(p − 1)

Note the following consequences for p = 2. 6.14.   1,     (−1)r ,       −1 2(−1)s , G2 (n) =    2(−1)r+s ,         0,

for n = 1, when 2 - n and n = p1 · · · pr , where p1 , . . . , pr are distinct primes, when n = 2s , s > 1, when 2 | n and n = 2s p1 · · · pr , s > 1, where p1 , . . . , pr are distinct odd primes, in other cases.



25

s 6.15. If 2 - n, then G−1 2 (n) = µ(n). If 2 | n and n = 2 m with 2 - m, then s G−1 2 (n) = (−1) 2µ(m).

All the values of G−1 2 belong to {−2, −1, 0, 1, 2}. Note some properties of the function Gc . Proposition 6.16 ([13]). (1) G2 ∗ G3 = G1 ∗ G6 . (2) G12 ∗ G18 = G6 ∗ G36 = µ ∗ µ ∗ G2 ∗ G3 ∗ G4 ∗ G9 . (3) Ga ∗ Gb = G(a,b) G[a,b] for a, b ∈ N. (4) If (a, b) = 1, then Gab = µ ∗ Ga ∗ Gb . n P (5) The function n 7→ Gc (n) is multiplicative. (Mathematical Olympiad, Italy 2005). c=1

We denote by Hc the arithmetic function defined by n 1 , for n ∈ N. Hc (n) = [c, n] = c (c, n) Proposition 6.17. The function Hc is multiplicative. Proof. Let a, b ∈ N, (a, b) = 1. Then Hc (ab) =

ab ab = = Hc (a)Hc (b). Gc (ab) Gc (a)Gc (b)

We applied Proposition 6.9. If c is a positive integer then we denote by εc the multiplicative function defined by ( 1, if n | c, εc (n) = 0, otherwise. It is easy to check that if n > 2 and (c, n) = 1, then ε−1 c (n) = 0. Using this fact and −1 the fact that the function εc is multiplicative, we obtain the following proposition. Proposition 6.18. n n (1) If p is a prime number and n ∈ N, then ε−1 p (p ) = (−1) . (2) If p is a prime number and s, n ∈ N, then   1, if n ≡ 0 (mod s + 1), −1 n −1, if n ≡ 1 (mod s + 1), εps (p ) =  0, otherwise. (3) Let c = pα1 1 · · · pαs s be the prime decomposition of c > 2, and let n = pi11 · · · piss ·m with (m, c) = 1. If m > 2, then εc−1 (n) = 0. If m = 1, then −1 −1 is α1 ε−1 (n) = ε pi11 · · · εpαs s p1 . c p1 Colorary 6.19. If 2 6 c = p1 · · · ps is a square-free positive integer, then ε−1 pi11 · · · piss = (−1)i1 +···+is , c and ε−1 pi11 · · · piss · m = 0 if m > 2 and (m, c) = 1, where i1 , . . . , is are nonnegative c integers.


6.2


26

Examples of factor-closed sets

A finite set S of positive integers is said to be factor-closed if all positive factors of any element of S belong to S, ([4, 22]). Example 6.20. All the following sets are factor-closed: (1) (2) (3) (4)

{1, 2, 3, . . . , n}; {1, 2, 3, 6}, {1, 2, 3, 4, 6, 8, 12, 24}, {1, 2, 3, 5, 6, 10, 15, 30}; {1, 3, 5, 7, . . . , 2n − 1}; {1, p1 , p2 , . . . , pn } where p1 , . . . , pn are distinct primes.

Proposition 6.21 ([4]). Let S be the arithmetic progression defined as follows S = {s, s + d, s + 2d, . . . , s + (n − 1)d} where (s, d) = 1. The set S is factor-closed if and only if (1) s = 1, d = 1; S = {1, 2, . . . , n}; or (2) s = 1, d = 2; S = {1, 3, 5, . . . , 2n − 1}; or (3) S = {1, p1 , p2 , . . . pn−1 }, where p1 , . . . , pn−1 are primes; for example: {1, 31, 61}, {1, 19, 37}, {1, 37, 73, 109}. A finite set {x1 , . . . , xn } is said to be gcd-closed if (xi , xj ) ∈ S for 1 6 i, j 6 n ([7], [19]). Clearly, a factor-closed set is gcd-closed but not conversely. For example, the set {2, 4, 6} is gcd-closed but not factor-closed ([19]). Proposition 6.22 ([6]). Let S be the arithmetic progression defined as follows S = {s, s + d, s + 2d, . . . , s + (n − 1)d} where (s, d) = 1. If n > 4, then the set S is gcd-closed if and only if exactly one of the following conditions is satisfied: (1) s = 1, d = 1; S = {1, 2, . . . , n}; (2) s = 1, d = 2; S = {1, 3, 5, . . . , 2n − 1}; (3) s = 1, d = 4; S = {1, 5, 9, . . . , 4n − 3} with n 6 5; (4) s = 1, d = 6; S = {1, 7, 13, . . . , 6n − 5} with n 6 9; (5) s = 1, d = 2k, k is odd and k > 5; S is a set of pairwise relatively prime numbers with n 6 9; (5) s = 1, d = 2k, k is even and k > 5; S is a set of pairwise relatively prime , where p is the smallest odd prime dividing k − 1. numbers with n 6 3p−1 2



27

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