Hankel determinants with arithmetic functions Andrzej Nowicki Toru´ n 18.06.2017
Contents 1 Introduction
1
2 Notations and preparatory facts
2
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 6 8 10 11 13 15
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
20
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 M¨obius 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
Andrzej Nowicki, 2017,
Hankel determinants with arithmetic functions
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.
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Hankel determinants with arithmetic functions
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.
Andrzej Nowicki, 2017,
Hankel determinants with arithmetic functions
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
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Hankel determinants with arithmetic functions
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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
Andrzej Nowicki, 2017,
Hankel determinants with arithmetic functions
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).
Andrzej Nowicki, 2017,
3.2
Hankel determinants with arithmetic functions
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:
Andrzej Nowicki, 2017,
Hankel determinants with arithmetic functions
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)
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3.3
Hankel determinants with arithmetic functions
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 )
Andrzej Nowicki, 2017,
3.18. Let Dn = det
Hankel determinants with arithmetic functions
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 )
Andrzej Nowicki, 2017,
Hankel determinants with arithmetic functions
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 M¨obius 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.
Andrzej Nowicki, 2017,
Hankel determinants with arithmetic functions
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)
Andrzej Nowicki, 2017,
Hankel determinants with arithmetic functions
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).
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3.6
Hankel determinants with arithmetic functions
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.
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Hankel determinants with arithmetic functions
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
Andrzej Nowicki, 2017,
4.1
Hankel determinants with arithmetic functions
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]).
Andrzej Nowicki, 2017,
4.2
Hankel determinants with arithmetic functions
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.
Andrzej Nowicki, 2017,
4.3
Hankel determinants with arithmetic functions
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.
Andrzej Nowicki, 2017,
Hankel determinants with arithmetic functions
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.
Andrzej Nowicki, 2017,
6
Hankel determinants with arithmetic functions
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.
Hankel determinants with arithmetic functions
([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
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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.
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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.
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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.
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6.2
Hankel determinants with arithmetic functions
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
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