with arithmetic functions

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with arithmetic functions

Andrzej Nowicki Toru´n 18.06.2017

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 b c d c d e

,

a b c d b c d e c d e f 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).

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In 1875, J. H. S. Smith [32] proved that

(1, 1) (1, 2) · · · (1, n) (2, 1) (2, 2) · · · (2, n)

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

= ϕ(1)ϕ(2) · · · ϕ(n),

where ϕ is the Euler totient function. He also showed that

[1, 1] [1, 2] · · · [1, n]

[2, 1] [2, 2] · · · [2, n]

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

= n!Y

p

(1 − p)^[n/p],

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 M¨obius function. We use also the standard functions I, T , T^r and e. If n is a positive integer, then

I(n) = 1, T (n) = n, T^r(n) = n^r, e(n) = 1 n

=

( 1, for n = 1 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

f ∗ g

(n) =X

d|n

f (d)g

n d

,

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

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[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⁻¹is 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 J_r(n) = n^rY

p|n

1 − 1

p^r

,

where p ranges over all prime divisors of n. The function J_r 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 J_r∗ I = T^r. This implies that

J_r = T^r∗ µ, J_r⁻¹ = µT^r∗ I.

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where J_r⁻¹ is the inverse of J_r with respect to Dirichlet convolution. It is clear that J_r(1)J_r(2) · · · J_r(n) = (n!)^rY

p

1 − 1

p^r

[n/p]

.

For any real r, we denote by π_rthe multiplicative function defined by π_r(p^m) = −p^r, for each prime power p^m. Thus, for n ∈ N, we have

π_r(n) =Y

p|n

(−p^r),

where p ranges over all prime divisors of n. We denote by δ_rthe multiplicative function π_rJ_r, that is, for all n ∈ N,

δ_r(n) = n^rY

p|n

(1 − p^r).

In particular, δ1(n) = ϕ(n)π1(n) = nQ

p|n

(1 − p).

Let c be a fixed positive integer. We denote by G_c the arithmetic function defined by

G_c(n) = gcd(c, n) = (c, n)

for all n ∈ N. In particular, G1 = I. The function G_cis multiplicative (see Proposition 6.9 in Appendix).

The arithmetic function n 7→ [c, n] is clearly not multiplicative. We denote by H_c the arithmetic function defined by

H_c(n) = 1

c[c, n] = n

(c, n), for n ∈ N.

This function is multiplicative (see Proposition 6.17). We consider also the next multiplicative function, denoted by ε_c, defined by

ε_c(n) =

( 1, if n | c, 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), ε_c(n) = 1 when n | c, and ε_c(n) = 0 when n - c. Observe that if a, b are coprime positive integers, then the numbers G_c(a) and G_c(b) are coprime, too. This implies that if f is a multiplicative function, then the function f ◦ G_c, 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 F_c= f ◦ G_c. Then F_c∗ µ = ε_cg, that is, if n ∈ N, then

(Fc∗ µ) (n) =

( g(n), if n | c, 0, otherwise.

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Proof. Put h := F_c∗ µ, and let n ∈ N.

Step 1. Assume that n | c. In this case h(n) = g(n). Indeed, h(n) = (Fc∗ µ) (n) =P

d|n

Fc(d)µ ⁿ_d = P

d|n

f

(c, d)

µ ⁿ_d

= P

d|n

f (d)µ ⁿ_d = (f ∗ µ) (n) =

(g ∗ I) ∗ µ (n)

=

g ∗ (I ∗ µ)

(n) = g(n).

Thus, if n | c then h(n) = ε_c(n)g(n) = (ε_cg) (n).

Step 2. Assume that n > 2 and (c, n) = 1. Then n - c. In this case h(n) = 0.

Indeed,

h(n) = (Fc∗ µ) (n) =P

d|n

Fc(d)µ ⁿ_d = P

d|n

f

(c, d)

µ ⁿ_d

= P

d|n

f (1)µ ⁿ_d = P

d|n

µ(d) = 0.

We applied the equality µ ∗ I = e. Thus, if n > 2 and (c, n) = 1, then h(n) = 0 = 0 · g(n) = ε_c(n)g(n) = (ε_cg) (n).

Step 3. Assume that n is a prime power p^α 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 = p^βu, p - u, 1 6 β < α. Then h(n) = (F_c∗ µ) (n) = P

d|p^α

F_c(d)µ ^p_d^α = Fc(p^α) µ(1) + F_c(p^α−1) µ(p)

= f

(c, p^α)

− f

(c, p^α−1)

= f

(p^βu, p^α)

− f

(p^βu, p^α−1)

= f p^β − f p^β = 0.

Thus, if n = p^α with α > 1, and c < n, then h(n) = 0 = 0·g(n) = εc(n)g(n) = (ε_cg) (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 = p^α₁¹· · · p^α_s^s. Since n - c, there exists an i ∈ {1, . . . , s} such that p^α_iⁱ - c. Put p := pi and α := α_i. Let c = up^α + v, where u, v ∈ Z and 1 6 v < p^α. Then h(p^α) = 0. Indeed, using Step 3, we have

h(p^α) = (Fc∗ µ) (p^α) = P

d|p^α

Fc(d)µ ^p_d^α = P

d|p^α

f

(up^α+ v, d)

µ ^p_d^α

= P

d|p^α

f (v, d)

µ ^p_d^α = P

d|p^α

F_v(d)µ ^p_d^α = (F_v ∗ µ) (p^α) = 0.

Hence, h (p^α_iⁱ) = 0. But the function h is multiplicative, so

h(n) = h (p₁^α¹· · · p^α_s^s) = h (p₁^α¹) · · · h (p^α_iⁱ) · · · h (p^α_s^s) = 0.

Thus, in any case we have the equality h(n) = (ε_cg) (n). This completes the proof. Proposition 3.2. Let f, g be multiplicative functions such that f = g ∗ I, and let S = {x₁, . . . , x_n} be a factor-closed set of positive integers. Assume that x₁ < x₂ < · · · < x_n. Then for every i ∈ {1, . . . , n} the following equalities hold.

X

xk|xn

f

(x_i, x_k) µ x_n

xk

=







g(x_n), if i = n, 0, otherwise.

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Proof. Let i ∈ {1, . . . , n}, and let F_x_i = f ◦ G_x_i, that is, F_x_i(m) = f

(x_i, m) for all m ∈ N. Then we have

X

x_k|xn

f

(x_i, x_k) µ x_n

x_k

= X

x_k|xn

F_x_i(x_k)µ x_n x_k

=X

d|xn

F_x_i(d)µx_n d

= (F_x_i ∗ µ) (x_n).

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, . . . , x_n} (where n > 1) be a factor-closed set of positive integers with x₁ < x₂ < · · · < x_n. If f, g : N → C are multiplicative functions such that f = g ∗ I, then

deth f

(x_i, x_j)i

16i,j6n

= g(x₁)g(x₂) · · · g(x_n).

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

d|m

g(d) for m ∈ N, then

f

(x₁, x₁) f

(x₁, x₂)

· · · f

(x₁, x_n) f

(x₂, x₁) f

(x₂, x₂)

· · · f

(x₂, x_n)

... ... ...

f

(x_n, x₁) f

(x_n, x₂)

· · · f

(x_n, x_n)

= g(x₁)g(x₂) · · · g(x_n).

Proof. (Smith [32]). Denote the matrix h f

(x_i, x_j)i

by M_n. Let D_n = det M_n, and let A₁, . . . , A_n be the columns of the matrix M_n. Replace the last column A_n by the sum

X

xk|xn

µ x_n x_k

A_k.

Then the value of D_n is not changed and, by Proposition 3.2, the last new column is equal to [0, 0, . . . , 0, g(x_n)]^T. Hence,

D_n= g(x_n)D_n−1 and consequently, D_n = g(x_n)g(x_n−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

deth f

(i, j)i

16i,j6n = g(1)g(2) · · · g(n).

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

d|m

g(d) for m ∈ N,

then

f

(1, 1)

f

(1, 2)

· · · f

(1, n)

f

(2, 1) f

(2, 2)

· · · f

(2, n)

... ... ...

f

(n, 1) f

(n, 2)

· · · 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 a_ij defined by

a_ij =

( 1, when j | i, 0, when j - i.

Then, for all i, j ∈ N, we have f i, j)

= X

d|(i,j)

g(d) =

n

X

k=1

a_ika_jkg(k), and this implies that D = A · B, where

A =

a₁₁ a₁₂ · · · a_1n a₂₁ a₂₂ · · · a_2n ... ... ... an1 an2 · · · ann

, B =

a₁₁g(1) a₂₁g(1) · · · a_n1g(1) a₁₂g(2) a₂₂g(2) · · · a_n2g(2)

... ... ...

a1ng(n) a2ng(n) · · · anng(n)

.

Observe that a_ii = 1 and a_ij = 0 for i < j. Hence, A = 1 and B = g(1)g(2) · · · g(n).

Therefore, D = g(1)g(2) · · · g(n).

Proposition 3.5. Let D_n = deth f

(i, j)i

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 D_n = g(1) · · · g(p) · · · g(n) = 0.

Example 3.6. Let D_n = deth f

(i, j)i

16i,j6n

, where f (x) = x² − 4x + 20. Then D₁ = 17, D₂ = −17, and D_n = 0 for n > 3. It follows from Proposition 3.3, because f (1) = f (3).

Example 3.7. Let D_n = deth f

(i, j)i

16i,j6n

, where f (x) = x² − 8x + 100. Then D_n= 0 for n > 7. It follows from Proposition 3.3, because f (1) = f (7).

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3.2 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) · · · (x₁, x_n) (x₂, x₁) (x₂, x₂) · · · (x₂, x_n)

... ... ...

(x_n, x₁) (x_n, x₂) · · · (x_n, x_n)

= ϕ(x₁)ϕ(x₂) · · · ϕ(x_n).

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) · · · (1, n) (2, 1) (2, 2) · · · (2, n)

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

= ϕ(1)ϕ(2) · · · ϕ(n).

Examples: D₁ = D₂ = 1, D₃ = 2, D₄ = 4, D₅ = 16, D₆ = 32, D₇ = 2⁶3, D₈ = 2⁸3, D₉ = 2⁹3², D₁₀= 2¹¹3², D₁₁ = 2¹²3²5, D₁₂= 2¹⁴3²5.

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, . . . , x_n} 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 [(x_i, x_j)] 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(x_n) then S is factor-closed. In 1990, Zhongshan Li [22]

proved that this conjecture is true.

Theorem 3.10 (Li [22]). Let S = {x₁, . . . , x_n} 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(x_n) holds if and only if the set S is factor-closed.

The matrix (S) is always positive definite ([4, 22]). Moreover (see [22]), det(S) 6 x¹x2· · · xn− 1

2n! . Note also:

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Proposition 3.11. Let S={x₁, . . . , x_n} be an ordered set of distinct positive integers.

If S is factor closed, then

ϕ

(x₁, x₁) ϕ

(x₁, x₂)

· · · ϕ

(x₁, x_n) ϕ

(x₂, x₁) ϕ

(x₂, x₂)

· · · ϕ

(x₂, x_n)

... ... ...

ϕ

(x_n, x₁) ϕ

(x_n, x₂)

· · · ϕ

(x_n, x_n)

= h(x₁)h(x₂) · · · h(x_n),

where h = ϕ ∗ µ.

Proof. This is an immediate consequence of Theorem 3.3, because ϕ = h ∗ I.

Example 3.12.

ϕ (1, 1)

ϕ (1, 2)

· · · ϕ

(1, n) ϕ

(2, 1) ϕ

(2, 2)

· · · ϕ

(2, n)

... ... ...

ϕ

(n, 1) ϕ

(n, 2)

· · · 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 (p^m) = 1 + p^m− 1

p^m−1(p − 1)²

for each prime power p^m. Let S = {x₁, . . . , x_n} be a factor-closed set of positive integers. Then

f

(x₁, x₁) f

(x₁, x₂)

· · · f

(x₁, x_n) f

(x₂, x₁) f

(x₂, x₂)

· · · f

(x₂, x_n)

... ... ...

f

(x_n, x₁) f

(x_n, x₂)

· · · f

(x_n, x_n)

= 1

ϕ(x1)ϕ(x2) · · · ϕ(xn).

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

p^m−1(p − 1) = 1 ϕ(p^m). Hence, g(m) = _ϕ(m)¹ 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 equal to g(x₁) · · · g(x_n) = _ϕ(x ¹

1)···ϕ(xn).

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3.3 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

Jr(n) = n^rY

p|n

1 − 1

p^r

for n ∈ N, where p ranges over all prime divisors of n. In particular, J1 = ϕ and J₀ = e.

We know that J_r∗ I = T^r, where T^r(n) = n^r for n ∈ N (see Proposition 6.7). This implies that J_r= T^r∗ µ, J_r⁻¹ = µT^r∗ I, where J_r⁻¹ is the inverse of J_r with respect to Dirichlet convolution.

Theorem 3.14 ([32, 11]). Let S={x₁, . . . , x_n} 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, x_n)^r

... ... ...

(x_n, x₁)^r (x_n, x₂)^r · · · (x_n, x_n)^r

= J_r(x₁)J_r(x₂) · · · J_r(x_n).

Proof. Use Theorem 3.3 and the equality J_r∗ 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 (2, 1)^r (2, 2)^r · · · (2, n)^r

... ... ...

(n, 1)^r (n, 2)^r · · · (n, n)^r

= J_r(1)J_r(2) · · · J_r(n) = (n!)^rY

p

1 − 1

p^r

[n/p]

,

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 D_n every determinant, which appears in the presented theorems and propositions.

3.16. Let D_n= det [(i, j)²]_n×n = det [(i², j²)]_n×n. Then D_n= J₂(1)J₂(2) · · · J₂(n) = (n!)²Y

p

1 − 1

p²

[n/p]

, where p ranges over all primes belonging to {1, 2, . . . , n}.

Examples: D₁ = 1, D₂ = 3, D₃ = 24, D₄ = 288 = 2⁵3², D₅ = 2⁸3³, D₆ = 2¹¹3⁴, D₇ = 2¹⁵3⁵, D₈ = 2¹⁹3⁶, D₁₁ = 2²⁸3¹¹5.

3.17. Let S={x₁, . . . , x_n} be an ordered set of distinct positive integers, and let r be a real number. If S is factor closed, then

p(x₁, x₁) p(x₁, x₂) · · · p(x₁, x_n) p(x2, x₁) p(x2, x₂) · · · p(x2, x_n)

... ... ...

p(xn, x1) p(xn, x2) · · · p(xn, xn)

= J_1/2(x₁)J_1/2(x₂) · · · J_1/2(x_n).

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3.18. Let D_n= deth

p(i, j)i

n×n. Then D_n =p(n!) Q

p

1 − ^√¹_p[n/p]

, where p ranges over all primes belonging to {1, 2, . . . , n}.

Proposition 3.19. Let S={x₁, . . . , x_n} be an ordered set of distinct positive integers, and let r be a real number. If S is factor closed, then

1 (x1,x1)

1

(x1,x2) · · · _(x¹

1,xn) 1

(x2,x1) 1

(x2,x2) · · · _(x¹

2,xn)

... ... ...

1 (xn,x1)

1

(xn,x2) · · · _(x ¹

n,xn)

= J−1(x₁)J−1(x₂) · · · J−1(x_n).

3.20. Let D_n= deth

1 (i,j)

i

n×n

. Then D_n= 1 n!

Y

p

(1 − p)^[n/p].

Examples: D₁ = 1, D₂ = −¹₂, D₃ = ¹₃, D₄ = −₁₂¹, D₅ = ₁₅¹, D₆ = ₄₅¹, D₇ = −₁₀₅² , D₈ = ₄₂₀¹ , D₉ = −₁₈₉₀¹ , D₁₀ = −₄₇₂₅¹ . ([34], see Theorem 3.15).

3.21. Let D_n= deth

1 (i,j)²

i

n×n

= deth

1 (i²,j²)

i

n×n

. Then D_n= 1 (n!)²

Y

p

1 − p²[n/p]

. Examples: D₁ = 1, D₂ = −³₄, D₃ = ²₃, D₄ = −₈¹, D₅ = ₂₅³, D₆ = ₂₅², D₇ = −₁₂₂₅⁹⁶ , D₈ = ₂₄₅₀⁹ , D₉ = −₁₁₀₂₅⁴ , D₁₀= −₃₀₆₂₅⁸ . (see Theorem 3.15).

3.22. If r is a real number, and {x1, . . . , xn} is a factor-closed set of positive integers, then

J_r

(x₁, x₁) J_r

(x₁, x₂)

· · · J_r

(x₁, x_n) J_r

(x₂, x₁) J_r

(x₂, x₂)

· · · J_r

(x₂, x_n)

... ... ...

J_r

(x_n, x₁) J_r

(x_n, x₂)

· · · J_r

(x_n,_n)

= h(x₁)h(x₂) · · · h(x_n),

where h = J_r∗ µ is the multiplicative function defined, for prime powers p^m, by h(p) = p^r− 2 and h(p^m) = p^mr

1 − _p¹r

2

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, x₁) τ

(x₁, x₂)

· · · τ

(x₁, x_n) τ

(x₂, x₁) τ

(x₂, x₂)

· · · τ

(x₂, x_n)

... ... ...

τ

(x_n, x₁) τ

(x_n, x₂)

· · · τ

(x_n, x_n)

= 1.

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

τ (2, 2)

τ (2, 3)

· · · τ

(2, n) τ

(3, 2) τ

(3, 3)

· · · τ

(3, n)

... ... ...

τ

(n, 2) τ

(n, 3)

· · · 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

D =







d22 d23 · · · d2n

d₃₂ d₃₃ · · · d_3n ... ... ... d_n2 d_n3 · · · d_nn







, E =







1 d₁₂ d₁₃ · · · d_1n 0 d₂₂ d₂₃ · · · d_2n 0 d₃₂ d₃₃ · · · d_3n ... ... ... 0 d_n2 d_n3 · · · d_nn





 .

Since det D = det E, we need to show that det E is equal to the number of all square- free 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 E_i, by

X

k|i

µ i k

E_k.

Let γ_ij = 1 if i | j, and γ_ij = 0 if i - j. Observe that P

k|i

γ_kj = d_ij. Hence, using the M¨obius inversion formula we obtainP

k|i

µ _kⁱ d_kj = γ_ij. This implies that det E = det F , where

F =







1 γ₁₂ γ₁₃ · · · γ_1n µ(2) γ₂₂ γ₂₃ · · · γ_2n µ(3) γ₃₂ γ₃₃ · · · γ_3n ... ... ... µ(n) γ_n2 γ_n3 · · · γ_nn





 .

Denote by F_i the i-th row of F . Let G = [g_ij]_16i,j6n be the matrix obtained from F by replacing F₁ by

n

P

i=1

µ(i)F_i. Then det G = det F = det D. If j > 1, then g_1j = 0. In fact,

g_1j =

n

X

i=1

µ(i)γ_ij =X

i|j

µ(i) = 0.

Moreover, g₁₁ =

n

P

i=1

µ(i)² is equal to the number of all square-free integers belonging to {1, 2, . . . , n}. Observe that [g_ij]_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.

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Proposition 3.25. Let {x₁, . . . , x_n} be a factor-closed set of positive integers.

If h = I ∗ I ∗ I = τ ∗ I, then

h

(x₁, x₁) h

(x₁, x₂)

· · · h

(x₁, x_n) h

(x₂, x₁) h

(x₂, x₂)

· · · h

(x₂, x_n)

... ... ...

h

(x_n, x₁) h

(x_n, x₂)

· · · h

(x_n, x_n)

= τ (x₁)τ (x₂) · · · τ (x_n).

3.5 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 σ_r(n) is the sum of the powers r of all positive divisors of n. In particular, σ₁ = σ.

Note that σ = T ∗ I and σ_r= T^r∗ I. Hence, immediately from Theorem 3.3 we obtain the following two propositions.

Proposition 3.26. If r is a real number and {x₁, . . . , x_n} is a factor-closed set of positive integers, then

σ_r

(x₁, x₁) σ_r

(x₁, x₂)

· · · σ_r

(x₁, x_n) σr

(x2, x1)

σr

(x2, x2)

· · · σr

(x2, xn)

... ... ...

σ_r

(x_n, x₁) σ_r

(x_n, x₂)

· · · σ_r

(x_n, x_n)

= (x₁x₂. . . x_n)^r.

Proposition 3.27. If {x₁, . . . , x_n} is a factor-closed set of positive integers, then

σ

(x₁, x₁) σ

(x₁, x₂)

· · · σ

(x₁, x_n) σ

(x₂, x₁) σ

(x₂, x₂)

· · · σ

(x₂, x_n)

... ... ...

σ

(x_n, x₁) σ

(x_n, x₂)

· · · σ

(x_n, x_n)

= x₁x₂. . . x_n.

Note some special cases of the above propositions, which appear in [32, 14, 9].

3.28.

σ (1, 1)

σ (1, 2)

· · · σ

(1, n) σ

(2, 1) σ

(2, 2)

· · · σ

(2, n)

... ... ...

σ

(n, 1) σ

(n, 2)

· · · σ

(n, n)

= n!.

3.29.

σ_r (1, 1)

σ_r (1, 2)

· · · σ_r

(1, n) σ_r

(2, 1) σ_r

(2, 2)

· · · σ_r

(2, n)

... ... ...

σ_r

(n, 1) σ_r

(n, 2)

· · · σ_r

(n, n)

= (n!)^r.

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3.30.

σ−1

(1, 1)

σ−1

(1, 2)

· · · σ−1

(1, n) σ₋₁

(2, 1)

σ₋₁ (2, 2)

· · · σ₋₁

(2, n)

... ... ...

σ−1

(n, 1) σ−1

(n, 2)

· · · σ−1

(n, n)

= 1 n!.

3.31. If f : N → C is the multiplicative function defined by f (p^m) = _p^pm^m+1(p−1)⁻¹ for each prime power p^m, then

f

(1, 1)

f

(1, 2)

· · · f

(1, n)

f

(2, 1) f

(2, 2)

· · · f

(2, n)

... ... ...

f

(n, 1) f

(n, 2)

· · · f

(n, n)

= 1 n!.

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

p^m(p − 1) − p^m− 1

p^m−1(p − 1) = 1 p^m.

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 number and {x₁, . . . , x_n} is a factor-closed set of positive integers, then

h

(x1, x1)

h

(x1, x2)

· · · h

(x1, xn)

h

(x₂, x₁) h

(x₂, x₂)

· · · h

(x₂, x_n)

... ... ...

h

(x_n, x₁) h

(x_n, x₂)

· · · h

(x_n, x_n)

= σ_r(x₁)σ_r(x₂) . . . σ_r(x_n),

where h = σ_r∗ I.

Proposition 3.33. If h = σ ∗ I, then

h (1, 1)

h (1, 2)

· · · h

(1, n) h

(2, 1) h

(2, 2)

· · · h

(2, n)

... ... ...

h

(n, 1) h

(n, 2)

· · · h

(n, n)

= σ(1)σ(2) · · · σ(n).

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3.6 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

e

(x1, x1)

e

(x1, x2)

· · · e

(x1, xn)

e

(x₂, x₁) e

(x₂, x₂)

· · · e

(x₂, x_n)

... ... ...

e

(x_n, x₁) e

(x_n, x₂)

· · · e

(x_n, x_n)

= µ(x1)µ(x2) . . . µ(xn).

Proposition 3.35.

e (1, 1)

e (1, 2)

· · · e

(1, n) e

(2, 1) e

(2, 2)

· · · e

(2, n)

... ... ...

e

(n, 1)

e

(n, 2)

· · · e

(n, n)

= µ(1)µ(2) · · · µ(n).

In particular: D₁ = 1, D₂ = −1, D₃ = 1, and D_n= 0 for n > 4.

µ

(x₁, x₁) µ

(x₁, x₂)

· · · µ

(x₁, x_n) µ

(x₂, x₁) µ

(x₂, x₂)

· · · µ

(x₂, x_n)

... ... ...

µ

(x_n, x₁) µ

(x_n, x₂)

· · · µ

(x_n, x_n)

= h(x₁)h(x₂) . . . h(x_n),

where h = µ ∗ µ = τ⁻¹. Colorary 3.37. Let

D_n=

µ (1, 1)

µ (1, 2)

· · · µ

(1, n) µ

(2, 1)

µ

(2, 2)

· · · µ

(2, n)

... ... ...

µ

(n, 1) µ

(n, 2)

· · · µ

(n, n) .

Then D₁ = 1, D₂ = −2, D₃ = 2², D₄ = 2², D₅ = −2³, D₆ = −2⁵, and D₇ = 2⁶, and D_n= 0 for n > 8.

In the next gcd-determinants appears the multiplicative function π_r. The following propositions are consequences of Theorem 3.3.

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Proposition 3.38. If r is a real number, and {x₁, . . . , x_n} is a factor-closed set of positive integers, then

π_r

(x₁, x₁) π_r

(x₁, x₂)

· · · π_r

(x₁, x_n) πr

(x2, x1)

πr

(x2, x2)

· · · πr

(x2, xn)

... ... ...

π_r

(x_n, x₁) π_r

(x_n, x₂)

· · · π_r

(x_n, x_n)

= h(x₁)h(x₂) . . . h(x_n),

where h is the multiplicative function defined, for each prime power p^m, by h(p^m) =

−(1 + p^r) for m = 1, and h(p^m) = 0 for m > 1.

Colorary 3.39.

π_r (1, 1)

π_r (1, 2)

· · · π_r

(1, n) πr

(2, 1)

πr

(2, 2)

· · · πr

(2, n)

... ... ...

π_r

(n, 1) π_r

(n, 2)

· · · π_r

(n, n)

= 0 for n > 4.

Proposition 3.40. Let r be a real number and let {x₁, . . . , x_n} be a factor-closed set of positive integers. Let f = πr ∗ I. Then f is the multiplicative function defined by f (p^m) = 1 − mp^r, for each prime power p^m, and we have

f

(x₁, x₁) f

(x₁, x₂)

· · · f

(x₁, x_n) f

(x₂, x₁) f

(x₂, x₂)

· · · f

(x₂, x_n)

... ... ...

f

(xn, x1)

f

(xn, x2)

· · · f

(xn, xn)

= πr(x1)πr(x2) · · · πr(xn).

Colorary 3.41.et r be a real number and let f = π_r∗ I. Then

f

(1, 1)

f

(1, 2)

· · · f

(1, n)

f

(2, 1) f

(2, 2)

· · · f

(2, n)

... ... ...

f

(n, 1) f

(n, 2)

· · · f

(n, n)

= π_r(1)π_r(2) · · · π_r(n) =Y

p

(−p^r)^[n/p].

4 Lcm-determinants

Let us recall (see Section 2) that if r is a real number, then π_r is the multiplicative function defined by π_r(p^m) = −p^r, for each prime power p^m. We denote by δ_r the multiplicative function J_rπ_r, that is, δ_r(n) = J_r(n)π_r(n) for n ∈ N. In particular,

δ₁(n) = ϕ(n)π₁(n) = nY

p|n

(1 − p), δ_r(n) = n^rY

p|n

(1 − p^r).

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4.1 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] · · · [1, n]

[2, 1] [2, 2] [2, 3] · · · [2, n]

[3, 1] [3, 2] [3, 3] · · · [3, n]

... ... ... ... [n, 1] [n, 2] [n, 3] · · · [n, n]

= δ₁(1)δ₁(2) · · · δ₁(n) = n!Y

p

(1 − p)^[n/p],

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.

Proof. ([34]). Let A, B, N be the n × n matrices defined by A = h [i, j]i

16i,j6n

, B = h

1 (i,j)

i

16i,j6n

and N = h n_iji

16i,j6n

, where n_ii = i and n_ij = 0 for i 6= j. Then A = N BN , det N = n!, and det B = _n!¹ Q

p

(1 − p)^[n/p] (see 3.20). Hence, det A = (n!)²det B = n!Y

p

(1 − p)^[n/p].

The equality

n

Q

k=1

δ1(k) = n!Q

p

(1 − p)^[n/p] is obvious.

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, . . . , x_n} be an ordered set of distinct positive integers. If S is factor-closed, then

[x₁, x₁] [x₁, x₂] [x₁, x₃] · · · [x₁, x_n] [x₂, x₁] [x₂, x₂] [x₂, x₃] · · · [x₂, x_n]

... ... ... ...

[x_n, x₁] [x_n, x₂] [x_n, x₃] · · · [x_n, x_n]

= δ₁(x₁)δ₂(x₂) · · · δ₁(x_n).

Proof. We do a small modification of the proof of Theorem 4.1. We use the equalities [x_i, x_j] = _(x^xⁱ^x^j

i,xj) 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

det[S] =

1 2 15 42

2 2 30 42

15 30 15 210 42 42 210 42

= 0.

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]).

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4.2 Power lcm-determinants

Theorem 4.3 ([32]). If r is a real number, then

[1, 1]^r [1, 2]^r · · · [1, n]^r [2, 1]^r [2, 2]^r · · · [2, n]^r [3, 1]^r [3, 2]^r · · · [3, n]^r

... ... ... [n, 1]^r [n, 2]^r · · · [n, n]^r

= δ_r(1)δ_r(2) · · · δ_r(n) = (n!)^rY

p

(1 − p^r)^[n/p],

where δ_r = π_rJ_r, and where p ranges over all primes belonging to the set {1, 2, . . . , n}.

Proof. Let A, B, N be the n × n matrices defined by A = h [i, j]^ri

, B = h

(i, j)^−ri

, N = h n_iji

, where N is the diagonal matrix with n_ii = i^r and n_ij = 0 for i 6= j. Then A = N BN , det N = (n!)^r, and B = (n!)^−rQ

p

(1 − p^r)^[n/p] (see Theorem 3.15). Hence, det A = (n!)^2rdet B = (n!)^rQ

p

(1 − p^r)^[n/p].

Colorary 4.4.

1 [1,1]

1

[1,2] · · · _[1,n]¹

1 [2,1]

1

[2,2] · · · _[2,n]¹ ... ... ...

1 [n,1]

1

[n,2] · · · _[n,n]¹

= δ−1(1)δ−1· · · δ−1(n) = 1 n!

Y

p

1 − 1

p

[n/p]

,

where δ₋₁= π₋₁J₋₁, and where p ranges over all primes belonging to the set {1, 2, . . . , n}.

Examples: D₁ = 1, D₂ = ¹₄, D₃ = ₁₈¹, D₄ = ₁₄₄¹ , D₅ = ₉₀₀¹ , D₆ = ₁₆₂₀₀¹ , D₇ = ₁₃₂₃₀₀¹ , D₈ = _2116800¹ . Observe that D_n> 0 for every n.

Note also

Theorem 4.5 ([32, 11]). Let S={x₁, . . . , x_n} 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, x_n]^r [x₂, x₁]^r [x₂, x₂]^r · · · [x₂, x_n]^r [x₃, x₁]^r [x₃, x₂]^r · · · [x₃, x_n]^r

... ... ...

[xn, x1]^r [xn, x2]^r · · · [xn, xn]^r

= δ_r(x₁)δ_r(x₂) · · · δ_r(x_n).

Proof. We do a small modification of the proof of Theorem 4.3. We use the equalities [x_i, x_j] = _(x^xⁱ^x^j

i,xj) and we apply results from the previous section.

with arithmetic functions