Arithmetic Function

基礎數論

數

數論 基 數 數 ,

數 . 基 , 數論

. 數論 ( ) ,

Silverman A Friendly Introduction to Number Theory (Prentice Hall, Third Edition

2006). .

v

Chapter 2

Arithmetic Function

數 , 數 . 數論

數 數, arithmetic function.

論 arithmetic function.

2.1. Multiplicative Arithmetic Functions

arithmetic function , arithmetic function ?

數 . 數 ,

multiplicative arithmetic function.

Definition 2.1.1. N C 數 arithmetic function. f :N → C arithmetic function a, b∈ N gcd(a, b) = 1 f (ab) = f (a) f (b), f

multiplicative arithmetic function.

arithmetic function f multiplicative , f (ab) = f (a) f (b)

. gcd(a, b) = 1 . f a, b∈ N

f (ab) = f (a) f (b), f completely multiplicative. completely multiplicative

arithmetic function , 數,

multiplicative arithmetic function.

multiplicative arithmetic function . Example 2.1.2. Möbius µ-function,

µ(n) =





1, n = 1;

0, 數 p p²|n;

(−1)^r, n = p1··· pr, p1, . . . , pr 數.

µ multiplicative. a, b∈ N gcd(a, b) = 1. a = 1 µ(a) = µ(1) = 1 µ(ab) = µ(b) = µ(a)µ(b). b = 1 µ(ab) = µ(a)µ(b).

a > 1 b > 1 . 數基 (Theorem 1.5.1) a, b

21

a = pⁿ₁¹··· pⁿr^r b = q^m₁¹···q^mt^t n_i, m_j 0 a, b

數 p_i qj . ni mj 1, n1≥ 2, p²₁|a

p²₁|ab, µ(a) = 0 µ(ab) = 0, µ(ab) = µ(a)µ(b). n1=··· = nr= 1 m₁=··· = mt= 1 . ab = p₁··· pr·q1···qt p₁, . . . , p_r, q₁, . . . , q_t

數 µ(ab) = (−1)^r+t. µ(a) = (−1)^r µ(b) = (−1)^t, µ(ab) = µ(a)µ(b).

µ multiplicative arithmetic function.

µ completely multiplicative. a = b = p, p 數

. µ(a) = µ(b) = 1 µ(ab) = 0, µ(ab) ̸= µ(a)µ(b).

arithmetic function f multiplicative , ,

gcd(a, b) = 1 數 a, b f (ab) = f (a) f (b), .

f multiplicative , a, b∈ N gcd(a, b) = 1 f (ab)̸= f (a) f (b) .

multiplicative arithmetic function 基 .

Proposition 2.1.3. f 0 multiplicative arithmetic function. f (1) = 1,

數 p t∈ N, f (p^t) n∈ N, f (n) .

Proof. f multiplicative gcd(1, 1) = 1, f (1) = f (1) f (1) f (1) = 1 f (1) = 0. f (1) = 0, n∈ N, gcd(n, 1) = 1, f (n) = f (n) f (1) = 0.

f 0 數, f 0 數 , f (1) = 1.

n∈ N, n = 1, 前 f (n) = f (1) = 1. n > 1, 數基

n = pⁿ₁¹··· pⁿr^r, p_i 數 n_i∈ N. f multiplicative gcd(pⁿ₁¹, pⁿ₂²··· pⁿr^r) = 1 f (n) = f (pⁿ₁¹pⁿ₂²··· pⁿ_r^r) = f (pⁿ₁¹) f (pⁿ₂²··· pⁿ_r^r). 數 f (n) =

f (pⁿ₁¹)··· f (pⁿr^r). f (pⁿ_iⁱ) f (n) . 

Proposition 2.1.3 f multiplicative arithmetic function,

數 p t∈ N f (p^t) f 數. 前

f multiplicative. multiplicative .

multiplicative arithmetic function

multiplicative arithmetic function. .

Lemma 2.1.4. a, b∈ N gcd(a, b) = 1. d ab 數, a

數 d₁ b 數 d₂ d = d1d2.

Proof. . d1|a d2|b d = d1d2,

.

. d|ab, d₁|a d₂|b d = d₁d₂ ?

d₁d₂= d d₁|a d₁ a d 數. , d₁ a, d

數, d2= d/d1 b. d1= gcd(a, d)

. d₂= d/d₁, d = d₁d₂ d₁|a. d₂|b.

2.1. Multiplicative Arithmetic Functions 23

d|ab (d/d₁)|(a/d1)b. d₁= gcd(a, d) gcd(a/d₁, d/d₁) = 1 (Corollary 1.1.8), Proposition 1.2.6(1) d/d1|b, d2|b.

. d|ab d₁, d^′₁, d₂, d₂^′ ∈ N d = d₁d₂, d₁|a d₂|b d = d₁^′d₂^′, d₁^′|a d₂^′|b, d1= d^′₁ d2= d₂^′. d1d2= d₁^′d₂^′, d1|d^′₁d^′₂.

d₁|a, d₂^′|b gcd(a, b) = 1, gcd(d₁, d₂^′) = 1. Proposition 1.2.6(1) d1|d^′₁. d₁^′|d1 d1, d₁^′ ∈ N d1= d₁^′, d2= d₂^′. 

Lemma 2.1.4 gcd(a, b) = 1 ,

gcd(a, b) = 1, ab 數 d₁|a, d2|b d = d₁d₂.

, gcd(a, b) = 1 . a = 6, b = 4 d = 6 ,

d₁= 6, d₂= 1 d₁^′ = 3, d₂^′ = 2 , .

a d 數 d1

. d₁ a, b 數 . ,

, .

Lemma 2.1.4 gcd(a, b) = 1 , d1, . . . , di, . . . , dr e1, . . . , ej, . . . , es

a b 數, d₁e₁, . . . , d_ie_j, . . . , d_re_s ab 數.

diej ab 數, Lemma 2.1.4 ab 數

diej diej .

multiplicative arithmetic function multiplicative arithmetic function.

Theorem 2.1.5. f :N → C multiplicative arithmetic function. 數

F :N → C n∈ N,

F(n) =

∑

d|n,d>0

f (d), F multiplicative arithmetic function.

Proof. F(n) =∑d|n,d>0 f (d) d₁, . . . , d_r n

數 F(n) = f (d1) +··· + f (dr). F multiplicative a, b∈ N

gcd(a, b) = 1 F(ab) = F(a)F(b).

d1, . . . , di, . . . dr e1, . . . , ej, . . . , es a b 數. F(a) = f (d₁) +···+ f (di) +···+ f (dr) F(b) = f (e₁) +···+ f (ej) +···+ f (es). F(a)F(b) = f (d1) f (e1) +··· + f (di) f (ej) +··· + f (dr) f (es). gcd(a, b) = 1 di, ej a, b 數, gcd(d_i, e_j) = 1. f multiplicative, d_i, e_j f (d_i) f (e_j) =

f (diej). F(a)F(b) = f (d1e1) +··· + f (diej) +··· + f (dres). Lemma 2.1.4 gcd(a, b) = 1, d₁e₁, . . . , d_ie_j, . . . , d_re_s ab 數,

F(ab) = F(a)F(b). 

Example 2.1.2 µ Theorem 2.1.5 multiplicative arithmetic function .

Example 2.1.6. δ : N → C arithmetic function , n∈ N, δ(n) =

∑

d|n,d>0µ(d),

µ möbius µ-function. µ multiplicative, Theorem 2.1.5 δ multi-

plicative. δ Proposition 2.1.3 δ(p^t) , p

數 t∈ N. p^t 數 1, p, p², . . . , p^t,

δ(p^t) =µ(1) + µ(p) + µ(p²) +··· +µ(p^t) = 1− 1 + 0 + ··· + 0 = 0.

n > 1, n = pⁿ₁¹··· pⁿr^r δ(n) = δ(pⁿ₁¹)···δ(pⁿr^r) = 0. δ(1) = µ(1) = 1,

δ(n) =

∑

d|n,d>0µ(d) =

{ 1, n = 1;

0, n > 1.

2.2. 數 數 數

multiplicative arithmetic function 數 數

數 數 .

數 n, v(n) n 數 數. n∈ N, v(n) ,

數 v :N → N. 數 , v arithmetic

function. n∈ N v(n) ? n 數 數

. 6 數 1, 2, 3, 6, v(6) = 4. ?

summation ∑ , v(n)

v(n) =

∑

d|n,d>0

1.

d d|n d > 0 , n 數

數.

Proposition 2.2.1. n∈ N, v(n) n 數 數. v :N → N

multiplicative arithmetic function. n = pⁿ₁¹··· pⁿr^r, p_i 數, v(n) = (n1+ 1)···(nr+ 1).

Proof. l:N → N arithmetic function n∈ N, l(n) = 1, v(n)

v(n) =

∑

d|n,d>0

l(d).

a, b∈ N, l(ab) = l(a)l(b) = 1, l (completely) multiplicative.

Theorem 2.1.5 v multiplicative.

v multiplicative, Proposition 2.1.3 n∈ N, v(n) .

數 p 數 t, v(p^t) . p^t 數 pⁱ,

2.2. 數 數 數 25

i∈ {0,1,...,t}, v(p^t) = t + 1. n∈ N, n = 1, v(n) = v(1) = 1;

n = pⁿ₁¹··· pⁿ_r^r pi 數, v multiplicative v(n) = v(pⁿ₁¹)···v(pⁿr^r) = (n1+ 1)···(nr+ 1).

 , 360 數 數, 360 = 2³· 3²· 5, Proposition 2.2.1,

v(360) = (3 + 1)(2 + 1)(1 + 1) = 24. multiplicative

arithmetic function . v(n)

. v multiplicative .

數 . 數 n, σ(n) n 數 .

n∈ N,σ(n) , 數 σ : N → N. 數

,σ arithmetic function. n∈ N σ(n) ? n

數 . 6 數 1, 2, 3, 6, σ(6) = 1+2+3+6 = 12.

? summation∑ , σ(n)

σ(n) =

∑

d|n,d>0

d.

d d|n d > 0 d, n 數 .

Proposition 2.2.2. n∈ N, σ(n) n 數 數. σ : N → N multiplicative arithmetic function. n = pⁿ₁¹··· pⁿ_r^r, pi 數,

σ(n) = pⁿ₁¹⁺¹− 1

p₁− 1 ···pⁿ_r^r+1− 1 p_r− 1 .

Proof. I : N → N arithmetic function n∈ N, I (n) = n, σ(n) σ(n) =

∑

d|n,d>0

I (d).

a, b∈ N, I (ab) = ab = I (a)I (b), I (completely) multiplicative.

Theorem 2.1.5 σ multiplicative.

σ multiplicative, Proposition 2.1.3 n∈ N, σ(n) .

數 p 數 t, σ(p^t) . p^t 數 pⁱ,

i∈ {0,1,...,t}, σ(p^t) = 1 + p +···+ p^t. 1, p, . . . , p^t p

數 ,

σ(p^t) = p^t+1− 1 p− 1 .

n∈ N, n = 1, σ(n) = σ(1) = 1; n = pⁿ₁¹··· pⁿr^r p_i 數, σ multiplicative

σ(n) = σ(pⁿ1¹)···σ(pⁿr^r) = pⁿ₁¹⁺¹− 1

p₁− 1 ···pⁿ_r^r+1− 1 p_r− 1 .



, 360 數 , 360 = 2³· 3²· 5, Proposition 2.2.2,

σ(360) = 2⁴− 1 2− 1

3³− 1 3− 1

5²− 1

5− 1 = 15· 13 · 6 = 1170.

2.3. The Euler ϕ^-function

n n 數 數.

Definition 2.3.1. n∈ N, ϕ(n) n n 數 數.

數 ϕ : N → N, Euler ϕ-function.

Euler ϕ-function multiplicative, 數 .

multiplicative arithmetic function f ϕ Theorem 2.1.5 , ϕ multiplicative. a, b∈ N gcd(a, b) = 1, ϕ(ab) = ϕ(a)ϕ(b).

a = 5, b = 4 . ϕ(20) = ϕ(5)ϕ(4). ϕ(20)

20 20 數 數, 20 數 :

1 6 11 16 2 7 12 17 3 8 13 18 4 9 14 19 5 10 15 20

5 10 15 20 數 5 數 20 ,

. 4 數 5 數 0 4

數 5 . 4 數 4 .

數 4 數 1 3 數 4 .

ϕ(5) = 4 數 5 , 4 數 ϕ(4) = 2 數 4 , 1

20 ϕ(5)ϕ(4) = 8 數 5 4 . 數 1 20 20

數, ϕ(20) = ϕ(5)ϕ(4).

前 . 前

數 20 , 數. a = 5, b = 4

數 20 . 20 數

5 4 數, .

Lemma 2.3.2. a, b, c∈ Z. gcd(ab, c) = 1 gcd(a, c) = 1 gcd(b, c) = 1.

Proof. gcd(ab, c) = 1. d = gcd(a, c), d a, c 數, d ab c

數, d = 1. gcd(b, c) = 1.

, gcd(a, c) = 1 gcd(b, c) = 1. gcd(ab, c)̸= 1, 數 p

p|gcd(ab,c). p|ab p|c. p 數, Lemma 1.4.2 p|a p|b. p

a, c b, c 數. gcd(a, c) = 1 gcd(b, c) = 1 , gcd(ab, c) = 1. 

2.3. The Eulerϕ-function 27

前 20 數 , 數 4 數 ,

gcd(a, b) = 1 .

Lemma 2.3.3. a, b, l∈ Z, b > 1 gcd(a, b) = 1. l, l + a, l + 2a, . . . , l + (b− 1)a,

數 b 數 . ϕ(b) b .

Proof. u, v∈ Z u, v b 數 , b|u − v. l, l + a, . . . , l + (b− 1)a

b 數 , l + ia, l + ja, 0≤ i < j ≤ b − 1,

b (l + ja)− (l + ia). b|(l + ja) − (l + ia), b|( j − i)a. gcd(a, b) = 1, Proposition 1.2.6(1) b| j − i. 0≤ i < j ≤ b − 1 , b

(l + ja)− (l + ia). l + ia, l + ja, 0≤ i < j ≤ b − 1, b

數 .

i∈ {0,1,...,b − 1} r_i l + ia b 數, 0≤ ri≤ b − 1 b

ri , {r0, r1, . . . , rb−1} {0,1,...,b − 1} . Lemma

1.3.1 gcd(l + ia, b) = gcd(r_i, b), {l,l + a,...,l + (b − 1)a} b 數

{0,1,...,b − 1} b 數 數 . {0,1,...,b − 1} ϕ(b) 數

b , . 

ϕ multiplicative arithmetic function.

Proposition 2.3.4. a, b∈ N gcd(a, b) = 1, ϕ(ab) = ϕ(a)ϕ(b).

Proof. ab 數 b :

1 1 + a ··· 1 + (b − 1)a 2 2 + a ··· 2 + (b − 1)a

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

a 2a ··· ba

l l, l + a, . . . , l + (b− 1)a. Lemma 1.3.1 數 a 數

l a 數 . 言 , l a l 數 a ; l

a l 數 a . 1≤ l ≤ a, ϕ(a) l a

. ϕ(a) 數 ( 數 a ab ).

ϕ(a) 數 a b . l, l + a, . . . , l + (b−1)a

, gcd(a, b) = 1 Lemma 2.3.3 ϕ(b) 數 b . 1 ab

ϕ(a)ϕ(b) a b . Lemma 2.3.2 數 ab 數.

ϕ(ab) = ϕ(a)ϕ(b). 

ϕ multiplicative, Proposition 2.1.3 ϕ .

Proposition 2.3.5. n = pⁿ₁¹··· pⁿ_r^r, pi 數, ϕ(n) = (pⁿ₁¹− pⁿ₁¹⁻¹)···(pⁿ_r^r− pⁿ_r^r−1) = n(1− 1

p1

)···(1 − 1 pr

).

Proof. 數 p 數 t,ϕ(p^t) . p p^t 數, u

p^t p u 數. p^t 數 p^t ,

數 p 數 . 1 p^t p^t/p 數 p 數.

1 pⁿ p^t− p^t−1 數 p^t .

n∈ N. n = 1, ϕ(n) = ϕ(1) = 1; n = pⁿ₁¹··· pⁿr^r p_i 數, ϕ multiplicative

ϕ(n) = ϕ(pⁿ1¹)···ϕ(pⁿr^r) = (pⁿ₁¹− pⁿ₁¹⁻¹)···(pⁿr^r− pⁿr^r−1) = n(1− 1 p1

)···(1 − 1 pr

).



ϕ multiplicative, Theorem 2.1.5 multiplicative

arithmetic function. F :N → N n∈ N, F(n) = ∑d|n,d>0ϕ(d). F multiplicative, 數 p t∈ N,

F(p^t) =ϕ(1) + ϕ(p) + ϕ(p²) +··· +ϕ(p^t) = 1 + (p− 1) + (p²− p) + ··· + (p^t− p^t−1) = p^t. .

Corollary 2.3.6 (Gauss). n∈ N

d|n,d>0

∑

Proof. F(n) =∑d|n,d>0ϕ(d), 前 F 0 數 F multiplicative, proposition 2.1.3 F(1) = 1. n∈ N n > 1 , n n = pⁿ₁¹··· pⁿ_r^r, p_i

數, F(p^t) = p^t Proposition 2.1.3

F(n) = F(pⁿ₁¹)···F(pⁿ_r^r) = pⁿ₁¹··· pⁿ_r^r = n,

. 

2.4. Convolution

convolution multiplicative arithmetic function, convo-

lution Möbius inversion formula.

, .

Definition 2.4.1. arithmetic functions f , g convolution f∗ g, n∈ N,

f∗ g(n) =

∑

d|n,d>0

f (d)g(n/d).

convolution , f∗ g(n) , n 數,

n 數 d, f (d)g(n/d) , . d n 數,

2.4. Convolution 29

e = n/d, de = n. , d, e 數 de = n, d|n.

f∗ g. ,

f∗ g(n) =

∑

d,e∈Nde=n

f (d)g(e).

數, d, e .

convolution .

f g arithmetic function, f∗ g 數 , f∗ g

arithmetic function. 言 convolution arithmetic function

( arithmetic function ).

基 .

Proposition 2.4.2. f , g, h arithmetic function. δ : N → N δ(n) =

{ 1, n = 1;

0, n > 1.

convolution .

(1) f∗δ = δ ∗ f = f.

(2) f∗ g = g ∗ f .

(3) ( f∗ g) ∗ h = f ∗ (g ∗ h).

Proof. (1) n∈ N, f ∗δ(n) = ∑d|n,d>0f (d)δ(n/d). n/d > 1 δ(n/d) =

0. ∑ , d = n , f∗δ(n) = f (n)δ(1) = f (n). 言 , f f∗δ

n∈ N . 數 , 數. δ ∗ f = f .

(2) n∈ N,

f∗ g(n) =

∑

d,ede=n∈N

f (d)g(e) =

∑

d,ede=n∈N

g(e) f (d) =

∑

d,ede=n∈N

g(d) f (e) = g∗ f (n).

f∗ g = g ∗ f .

(3) , n∈ N,

( f∗ g) ∗ h(n) =

∑

de=n d,e∈N

( f∗ g)(d)h(e)

=

∑

d,ede=n∈N

(

∑

r,srs=d∈N

f (r)g(s) )

h(e)

= _rse=n

∑

r,s,e∈N

f (r)g(s)h(e).

f∗ (g ∗ h)(n) =

∑

duv=n d,u,v∈N

f (d)g(u)h(v).

( f∗ g) ∗ h = f ∗ (g ∗ h). 

Proposition 2.4.2 , ∗ arithmetic function , δ

arithmetic function 1 ( , identity). ∗

. ∗ multiplicative arithmetic function

. .

Theorem 2.4.3. f , g multiplicative arithmetic function, f∗g multiplica- tive arithmetic function.

Proof. a, b∈ N gcd(a, b) = 1, f∗ g(ab) = ( f ∗ g(a))( f ∗ g(b)).

d, e∈ N de = ab, d|ab e|ab. Lemma 2.1.4 d₁, d₂

e1, e₂ d = d1d2 e = e1e2 d1, e₁ a 數 d2, e₂ b 數.

gcd(a, b) = 1, gcd(d1, d2) = 1 gcd(e1, e2) = 1. f , g multiplicative f∗ g(ab) =

∑

de=ab d,e∈N

f (d)g(e) =

∑

d1d2e1e2=ab d1|a,d2|b,e1|a,e2|b

d1,d₂,e₁,e₂∈N

f (d1) f (d2)g(e1)g(e2).

d₁, d₂, e₁, e₂ ∈ N d₁d₂e₁e₂ = ab d₁, e₁ d₂, e₂ a b 數. d1e1|ab, gcd(a, b) = 1 d1, e₁ a 數, gcd(d1e1, b) = 1.

Proposition 1.2.6(1) d1e1|a. a|d1e1d2e2, gcd(a, b) = 1 d2, e2 b 數, gcd(a, d₂e₂) = 1. a|d1e₁. a = d₁e₁, b = d₂e₂. , d1, d2, e1, e2∈ N a = d1e1 b = d2e2, d1d2e1e2= ab d1, e1 d2, e2

a b 數.

d1d2e

∑

d1,d2,e1,e2∈N

f (d₁) f (d₂)g(e₁)g(e₂) =

∑

d1e1=a,d2e2=b d1,d2,e1,e2∈N

f (d₁) f (d₂)g(e₁)g(e₂).

( f∗ g(a))( f ∗ g(b)) =

∑

d1e1=a d1,e1∈N

f (d1)g(e1)

∑

d2e2=b d2,e2∈N

f (d2)g(e2).

d1

∑

f (d1)g(e1)

∑

d2e2=b d2,e2∈N

f (d2)g(e2) =

∑

d1e1=a,d2e2=b d1,d2,e1,e2∈N

f (d1) f (d2)g(e1)g(e2).

. 

l:N → N arithmetic function n∈ N, l(n) = 1, arithmetic function f , n∈ N ,

f∗ l(n) =

∑

d,e∈Nde=n

f (d)l(e) =

∑

d|n,d>0

f (d).

l multiplicative arithmetic function, Theorem 2.1.5 Theo-

rem 2.4.3 .

2.5. The Möbius Inversion Formula 31

Example 2.4.4. Theorem 2.4.3 n∈ N,

d|n,d>0

∑

I : N → N arithmetic function n∈ N, I (n) = n. F :N → N

arithmetic function n∈ N, F(n) =

∑

d|n,d>0µ(d)n

d =

∑

d|n,d>0µ(d)I (n d).

F =µ ∗ I . µ I multiplicative, Theorem 2.4.3 F

multiplicative. 數 p t∈ N, F(p^t) .

µ(1) = 1, µ(p) = −1 i > 1 µ(pⁱ) = 0,

F(p^t) =µ(1)I (p^t) +µ(p)I (p^t−1) = p^t− p^t−1.

ϕ(p^t) ( Proposition 2.3.5), F ϕ multiplicative

Proposition 2.1.3 F =ϕ. n∈ N

d|n,d>0

∑

d =ϕ(n).

2.5. The Möbius Inversion Formula

前 Euler’s ϕ-function , arithmetic function f

ϕ-function ϕ(n) = ∑d|n,d>0f (d) . Möbius inversion formula f .

Theorem 2.5.1 (Möbius Inversion Formula). F, f arithmetic function, µ möbius µ-function. n∈ N, F, f

F(n) =

∑

d|n,d>0

f (d), n∈ N, F, f

f (n) =

∑

d|n,d>0

F(d)µ(n d).

Proof. l:N → N arithmetic function n∈ N, l(n) = 1. convolution F = f∗ l f = F∗µ.

F = f∗ l, F∗µ = ( f ∗ l) ∗ µ. Proposition 2.4.2(3) F∗µ = f ∗ (l ∗ µ).

n∈ N, l ∗µ(n) = µ ∗ l(n) = ∑d|n,d>0µ(d), Example 2.1.6 l∗µ = µ ∗ l = δ, δ : N → N

δ(n) =

{ 1, n = 1;

0, n > 1.

言 , F∗µ = f ∗ (l ∗ µ) = f ∗ δ. Proposition 2.4.2(1) F∗µ = f .

, f = F∗µ, f∗l = (F ∗µ)∗l = F ∗(µ ∗l). µ ∗l = δ f∗l = F ∗δ =

F. 

Möbius inversion formula n∈ N .

F(6) = f (1) + f (2) + f (3) + f (6) 論

f (6) = F(1)µ(6) + F(2)µ(3) + F(3)µ(2) + F(6)µ(1) = F(1) − F(2) − F(3) + F(6).

n∈ N 論 ( F(1) = f (1) , F(2) =

f (1) + f (2) F(3) = f (1) + f (3)).

Example 2.5.2. Möbius inversion formula, f ϕ(n) =

∑d|n,d>0 f (d). Möbius inversion formula f =µ ∗ϕ. µ ϕ multiplicative,

Theorem 2.4.3 f multiplicative. 數 p t∈ N, f (p^t)

.

f (p^t) =

∑

d|p^t,d>0

µ(d)ϕ(p^t

d) =µ(1)ϕ(p^t) +µ(p)ϕ(p^t−1) =ϕ(p^t)−ϕ(p^t−1).

f (p) = p−1−1 = p−2 t≥ 2 f (p^t) = p^t− p^t−1−(p^t−1− p^t−2) = p^t−2(p−1)². n = pⁿ₁¹··· pⁿr^r, p_i 數, f (n) = f (pⁿ₁¹)··· f (pⁿr^r).

f ( n_i= 1 ). Möbius inversion

formula, f µ(n) = ∑d|n,d>0 f (d). ϕ

multiplicative , Theorem 2.1.5 .

Example 2.5.2 arithmetic function F

arithmetic function f n∈ N, F(n) =∑d|n,d>0f (d). f multiplicative , Theorem 2.1.5 F multiplicative. , Corollary

F multiplicative, f multiplicative.

Corollary 2.5.3. F, f arithmetic function. n∈ N, F(n) =

∑

d|n,d>0

f (d)

F multiplicative arithmetic function, f multiplicative arithmetic function.

Proof. Theorem 2.5.1 f =µ ∗ F, µ multiplicative F multiplicative

, Theorem 2.4.3 f =µ ∗ F multiplicative. 

Example 2.5.4. 前 multiplicative arithmetic function

, Möbius inversion formula .

2.5. The Möbius Inversion Formula 33

(1) v(n) n 數 數. n∈ N,

v(n) =

∑

d|n,d>0

1 =

∑

d|n,d>0

l(d),

n∈ N, l(n) = 1. Möbius inversion formula n∈ N, 1 = l(n) =

∑

d|n,d>0µ(d)v(n

d) =

∑

d|n,d>0

v(d)µ(n d).

(2) σ(n) n 數 . n∈ N

σ(n) =

∑

d|n,d>0

d =

∑

d|n,d>0

I (d),

n∈ N, I (n) = n. Möbius inversion formula n∈ N, n =I (n) =

∑

d|n,d>0µ(d)σ(n

d) =

∑

d|n,d>0σ(d)µ(n d).

(3) Corollary 2.3.6 n∈ N

n =I (n) =

∑

d|n,d>0ϕ(d).

Möbius inversion formula n∈ N, ϕ(n) =

∑

d|n,d>0µ(d)I (n

d) =

∑

d|n,d>0µ(d)n d.

Example 2.5.4(3) 前 Example 2.4.4 multiplicative

. Example 2.5.4 multiplicative .

Möbius inversion formula multiplicative , arithmetic

function .