Download Arithmetic Functions: Identity, Unit, Divisor Power Sum, and Von Mangoldt Functions and more Lecture notes Law in PDF only on Docsity!
CHAPTER 2
Arithmetic Functions
2.1. Examples In this chapter, we discuss some basic arithmetic functions.
Definition 2.1. A real or complex valued function defined on the positive integers (or all integers) is called an arithmetic function or a number-theoretic function.
We give some examples of arithmetic functions as follows and we will discuss their properties in the following sections.
Example 2.2. We have the following simple but important arithmetic functions:
The identity function I(n) =
1 , if n = 1, 0 , otherwise, the unit function u(n) ≡ 1 , n ≥ 1 , N s(n) = ns, n ≥ 1 , s ∈ C.
Definition 2.3. The divisor function τ (n) is defined as the number of positive divisors of n, i.e.,
τ (n) =
d|n
Definition 2.4. The divisor power sum function σs(n) (with s ∈ C) is defined as the sum of s power of all positive divisors of n, i.e.,
σs(n) =
d|n
ds. (2.2)
Definition 2.5. The Euler totient function ϕ is defined as
ϕ(n) =
1 ≤k≤n (k,n)=
Definition 2.6. The M¨obius function μ is defined as follows:
μ(n) =
1 , if n = 1, (−1)r, if n = p 1 p 2 · · · pr, with distint primes pi, 0 , otherwise.
Definition 2.7. The von Mangoldt function Λ is defined as follows:
Λ(n) =
log p, if n = pk, k ≥ 1 and p prime, 0 , otherwise. (2.4)
7
8 2. ARITHMETIC FUNCTIONS
Definition 2.8. The omega function ω(n) is defined as the number of distinct prime factors of n, i.e.,
ω(n) = r, n = pa 11 pa 22 · · · pa r r is the prime-power decomposition. (2.5)
Definition 2.9. The Omega function Ω(n) is defined as the total number of prime factors of n, i.e.,
Ω(n) = a 1 + a 2 + · · · + ar, n = pa 11 pa 22 · · · pa r ris the prime-power decomposition. (2.6)
Definition 2.10. The Liouville function λ is defined as follows:
λ(n) = (−1)Ω(n). (2.7)
2.2. Multiplicative functions An important class of arithmetic functions are multiplicative functions defined as follows.
Definition 2.11. An arithmetic function f which is not identically zero is said to be multiplicative if
f (mn) = f (m)f (n) (2.8)
whenever (m, n) = 1. Moreover, if (2.8) holds for all m, n, then f is called completely multiplicative.
We have the following property of all multiplicative functions
Theorem 2.12. If f is multiplicative then f (1) = 1.
Proof. Since f is not identically zero, there exists n ∈ N such that f (n) 6 = 0. We have f (n) = f (n)f (1) as f is multiplicative. Hence f (1) = 1.
In this section, we will discuss some properties of some basic examples of multi- plicative functions.
2.2.1. The divisor function τ.
Theorem 2.13. If n = pa 11 pa 22 · · · pa r r, then
τ (n) =
∏^ r
i=
(ai + 1).
Proof. By Corollary 1.8 we have
τ (n) =
0 ≤c 1 ≤a 1
0 ≤c 2 ≤a 2
0 ≤cr ≤ar
∏^ r
i=
(ai + 1).
As a simple consequence we have the following corollary.
Corollary 2.14. The function τ is multiplicative.
10 2. ARITHMETIC FUNCTIONS
2.2.3. The M¨obius function μ. Note that μ(n) = 0 if and only if n has a square factor > 1. Here is a short table of values of μ:
n: 1 2 3 4 5 6 7 8 9 10 μ(n): 1 − 1 − 1 0 − 1 1 − 1 0 0 1 It is easy to prove the following property of μ.
Theorem 2.17. The function μ is multiplicative.
The M¨obius function arises in many different places in number theory. One of its fundamental properties is a remarkably simple formula for the divisor sum
d|n μ(d). Theorem 2.18. If n ≥ 1 , then we have ∑
d|n
μ(d) = I(n) =
1 , if n = 1, 0 , otherwise.
Proof. If n = 1, then both sides are equal to 1. If n > 1, then we can write n = pa 11 pa 22 · · · pa r r. By Corollary 1.8 and Theorem 2.17, we have ∑
d|n
μ(d) =
0 ≤c 1 ≤a 1
0 ≤c 2 ≤a 2
0 ≤cr ≤ar
μ(pc 11 pc 22 · · · pc rr )
0 ≤c 1 ≤ 1
0 ≤c 2 ≤ 1
0 ≤cr ≤ 1
μ(pc 11 )μ(pc 22 ) · · · μ(pc rr )
∏^ r
i=
0 ≤ci≤ 1
μ(pc i i) =
∏^ r
i=
This proves the theorem.
Theorem 2.19. If n ≥ 1 , then we have ∑
d^2 |n
μ(d) = |μ|(n) = μ(n)^2.
Proof. If n = 1, then both sides are equal to 1. If n > 1, then we can write n = pa 11 pa 22 · · · pa r r. By Corollary 1.8 and Theorem 2.17, we have ∑
d^2 |n
μ(d) =
0 ≤c 1 ≤[ a 21 ]
0 ≤c 2 ≤[ a 22 ]
0 ≤cr ≤[ a 22 ]
μ(pc 11 pc 22 · · · pc rr )
0 ≤c 1 ≤[ a 21 ]
0 ≤c 2 ≤[ a 22 ]
0 ≤cr ≤[ a 22 ]
μ(pc 11 )μ(pc 22 ) · · · μ(pc rr )
∏^ r
i=
0 ≤ci≤[ a 2 i ]
μ(pc ii ).
If there exists i such that ai ≥ 2 then
0 ≤ci≤[ a 2 i ] μ(p
ci i ) = 1^ −^ 1 = 0. Otherwise^ ai^ = 1 for all i, and hence
d^2 |n μ(d) =^ |μ|(n) = 1. This proves the theorem.^
2.2. MULTIPLICATIVE FUNCTIONS 11
2.2.4. Euler’s totient function ϕ. Note that ϕ(p) = p − 1 if p is prime. Here is a short table of values of ϕ:
n: 1 2 3 4 5 6 7 8 9 10 ϕ(n): 1 1 2 2 4 2 6 4 6 4 Theorem 2.20. If n ≥ 1 , then we have
ϕ(n) =
d|n
μ(d)
n d
Proof. By Theorem 2.18, we have ϕ(n) =
1 ≤k≤n (k,n)=
1 ≤k≤n
d|(n,k)
μ(d).
Exchanging the order of the sums above, we get
ϕ(n) =
d|n
μ(d)
1 ≤k≤n d|k
d|n
μ(d)
n d
as claimed.
Theorem 2.21. If n ≥ 1 , then we have
ϕ(n) = n
p|n
p
Proof. Assume n = pa 1 1 pa 2 2 · · · pa r r. By Corollary 1.8 and Theorems 2.17 and 2.20, we have
ϕ(n) = n
d|n
μ(d) d
= n
0 ≤c 1 ≤a 1
0 ≤c 2 ≤a 2
0 ≤cr ≤ar
μ(pc 11 pc 22 · · · pc rr ) pc 11 pc 22 · · · pc rr
= n
0 ≤c 1 ≤ 1
0 ≤c 2 ≤ 1
0 ≤cr ≤ 1
μ(pc 11 pc 22 · · · pc rr ) pc 11 pc 22 · · · pc rr
∏^ r
i=
0 ≤ci≤ 1
μ(pc i i) pc ii
= n
∏^ r
i=
pi
= n
p|n
p
Theorem 2.22. The function ϕ is multiplicative. Proof. For any m, n ∈ N such that (m, n) = 1, we need to prove ϕ(mn) = ϕ(m)ϕ(n). Assume m = pa 11 pa 22 · · · pa r r and n = q 1 b^1 qb 22 · · · qb ss , with pi, qj are distinct primes and ai, bj ∈ Z≥ 0. By Theorem 2.21 we have
ϕ(mn) = mn
p|mn
p
= m
∏^ r
i=
pi
n
∏^ s
j=
qj
= ϕ(m)ϕ(n).
This completes the proof.
Theorem 2.23. If n ≥ 1 , then we have n =
d|n
ϕ(d).
2.3. DIRICHLET CONVOLUTION 13
2.3.2. M¨obius transform. Definition 2.27. We define the M¨obius transform of an arithmetic function f to be F = f ∗ u, that is,
F (n) =
d|n
f (d).
Theorem 2.28 (M¨obius inversion formula). If F = f ∗ u, then f = F ∗ μ. Conversely, if f = F ∗ μ, then F = f ∗ u. We say f is the inverse M¨obius transform of F.
Proof. If F = f ∗ u, then by Theorem 2.18 and 2.26 we have F ∗ μ = (f ∗ u) ∗ μ = f ∗ (u ∗ μ) = f ∗ I = f. Conversely, if f = F ∗ μ, then f ∗ u = F ∗ μ ∗ u = F.
2.3.3. Dirichlet inverse. Theorem 2.29. If f is an arithmetic function with f (1) 6 = 0, then there is a unique arithmetic function f −^1 , called the Dirichlet inverse of f , such that
f ∗ f −^1 = f −^1 ∗ f = I.
Moreover, f −^1 is given by the recursion formulas
f −^1 (1) =
f (1)
, f −^1 (n) =
f (1)
d|n d<n
f
(n
d
f −^1 (d) for n > 1.
Proof. It is clear that the function f −^1 constructed above satisfies that f ∗f −^1 = f −^1 ∗ f = I. So we prove the existence. Now we prove the uniqueness. Assume that g is a function such that f ∗ g = g ∗ f = I. Then we have g = g ∗ I = g ∗ (f ∗ f −^1 ) = (g ∗ f ) ∗ f −^1 = I ∗ f −^1 = f −^1. This proves the theorem.
Remark 2.30. We have (f ∗ g)(1) = f (1)g(1). So if f (1) 6 = 0, g(1) 6 = 0, then (f ∗ g)(1) 6 = 0. The set of all arithmetic functions f with f (1) 6 = 0 forms an abelian group with respect to the operation ∗. Since (f −^1 ∗ g−^1 ) ∗ (f ∗ g) = I, we get (f ∗ g)−^1 = f −^1 ∗ g−^1 , if f (1) 6 = 0, g(1) 6 = 0.
Example 2.31. Recall that u ∗ μ = I. We have u−^1 = μ and μ−^1 = u. Theorem 2.32. Let f be multiplicative. Then f is completely multiplicative if and only if f −^1 (n) = μ(n)f (n), for all n ≥ 1. Proof. If f is completely multiplicative, then we have (μf ∗ f )(n) =
ab=n
μ(a)f (a)f (b) = f (n)
d|n
μ(d) = I(n).
So f −^1 = μf. If f −^1 = μf , then by Theorem 2.29 we have f (n) = −
aba<n=n
f −^1 (b) f (a) = −
aba<n=n
μ(b)f (b) f (a), for n > 1.
Let p be a prime and k ≥ 1 be an integer. Then we have
f (pk) = f (p)f (pk−^1 ).
14 2. ARITHMETIC FUNCTIONS
Hence f (pk) = f (p)k. This shows that f is completely multiplicative.
Theorem 2.33. We have λ−^1 = |μ|.
Proof. Note that λ is completely multiplicative. So we have λ−^1 = μλ. Note that μλ = |μ|. So we have λ−^1 = |μ|.
Theorem 2.34. We have ϕ−^1 = u ∗ μN.
Proof. Note that we have ϕ = μ ∗ N. By Theorem 2.32 we have N −^1 = μN. Hence ϕ−^1 = μ−^1 ∗ N −^1 = u ∗ μN.
Theorem 2.35. We have σ− s 1 = μN s^ ∗ μ, where N s(n) = ns.
Proof. Note that σs = u ∗ N s. By Theorem 2.32 we have (N s)−^1 = μN s. Hence σ−^1 = u−^1 ∗ (N s)−^1 = μ ∗ μN s.
2.3.4. Dirichlet convolution and multiplicative functions.
Theorem 2.36. If f and g are multiplicative, so is their Dirichlet convolution f ∗ g.
Proof. By corollary 1.8, for any pair of positive integers m, n with (m, n) = 1, a divisor d of mn can be written uniquely as a product of a divisor d 1 of m and a divisor d 2 of n. So
(f ∗ g)(mn) =
d|mn
f (d)g(mn/d) =
d 1 |m
d 2 |n
f (d 1 d 2 )g(mn/d 1 d 2 ).
Note that (d 1 , d 2 ) = (m/d 1 , n/d 2 ) = 1. Since f and g are multiplicative, we have
(f ∗ g)(mn) =
d 1 |m
f (d 1 )g(m/d 1 )
d 2 |n
f (d 2 )g(n/d 2 ) = (f ∗ g)(m)(f ∗ g)(n).
So f ∗ g is multiplicative.
Theorem 2.37. If both g and f ∗g are multiplicative, then f is also multiplicative.
Proof. We should assume that f is not multiplicative and then find a contra- diction. Assume that a pair of coprime positive integers (m, n) such that f (mn) 6 = f (m)f (n) and the product mn is smallest. Now we consider (f ∗ g)(mn). On the one hand, we have
(f ∗ g)(mn) =
d|mn
f (d)g(mn/d) =
d 1 |m
d 2 |n
f (d 1 d 2 )g(mn/d 1 d 2 ).
If d 1 d 2 < mn then we have f (d 1 d 2 ) = f (d 1 )f (d 2 ). Since g is multiplicative, we get
(f ∗ g)(mn) =
d 1 |m d 2 |n d 1 d 2 <mn
f (d 1 )f (d 2 )g(m/d 1 )g(n/d 2 ) + f (mn). (2.9)
16 2. ARITHMETIC FUNCTIONS
2.4. Generalized convolutions In this section, F denotes a real or complex-valued function defined on the positive real axis (0, +∞) such that F (x) = 0 for 0 < x < 1. Let g be an arithmetic function. Sums of the type (^) ∑
n≤x
g(n)F
( (^) x
n
arise frequently in number theory. The sum defines a new function H on (0, +∞) such that G(x) = 0 for 0 < x < 1. We denote the function H by g ◦ F , that is,
H(x) = (g ◦ F )(x) =
n≤x
g(n)F
( (^) x n
If F (x) = 0 for all non-integer x, the restriction of F to the integers is an arithmetic function and we have (g ◦ F )(m) = (g ∗ F )(m),
for all integers m ≥ 1. So the operation ◦ can be regarded as a generalization fo the Dirichlet convolution ∗.
Theorem 2.42. For any arithmetic functions f and g, we have f ◦ (g ◦ H) = (f ∗ g) ◦ H.
Proof. We have
(f ◦ (g ◦ H))(x) =
a≤x
f (a)
b≤x/a
g(b)H
( (^) x ab
n≤x
ab=n
f (a)g(b)
H
( (^) x n
= ((f ∗ g) ◦ H)(x).
This proves the theorem.
Theorem 2.43. If g has a Dirichlet inverse g−^1 , then the equation
H(x) =
n≤x
g(n)F
( (^) x n
implies
F (x) =
n≤x
g−^1 (n)H
( (^) x
n
Conversely, (2.14) implies (2.13).
Proof. If H = g ◦ F , then by Theorem 2.42 we have g−^1 ◦ H = g−^1 ◦ (g ◦ F ) = I ◦ F = F.
Conversely, if F = g−^1 ◦ H, then we have g ◦ F = g ◦ (g−^1 ◦ H) = I ◦ H = H.
Corollary 2.44. If g has a completely multiplicative function, then the equation
H(x) =
n≤x
g(n)F
( (^) x
n
2.5. THE VON MANGOLDT FUNCTION 17
if and only if
F (x) =
n≤x
μ(n)g(n)H
( (^) x n
Proof. By Theorem 2.32 we have g−^1 = μg.
2.5. The von Mangoldt function The von Mangoldt function Λ plays a central role in the distribution of primes. Here is a table of values of Λ(n):
n: 1 2 3 4 5 6 7 8 9 10 Λ(n): 0 log 2 log 3 log 2 log 5 0 log 7 log 2 log 3 0 Theorem 2.45. If n ≥ 1 then we have log n =
d|n
Λ(d).
That is, we have log = u ∗ Λ.
Proof. If n = pa 11 pa 22 · · · pa r r, then we have ∑
d|n
Λ(d) =
∑^ r
i=
∑^ ai
k=
Λ(pki ) =
∑^ r
i=
∑^ ai
k=
log pi =
∑^ r
i=
log pa i i= log n
as claimed.
Theorem 2.46. We have Λ(n) =
d|n
μ(d) log
n d
d|n
μ(d) log d.
That is we have Λ = μ ∗ log = −u ∗ μ log.
Proof. By Theorems 2.37 and 2.45, we have Λ = μ ∗ log. Note that ∑
d|n
μ(d) log
n d
= log n
d|n
μ(d) −
d|n
μ(d) log d = −
d|n
μ(d) log d.
Hence Λ = −u ∗ μ log. This completes the proof.
Definition 2.47. For k ≥ 0, the von Mangoldt function of degree k is defined as follows
Λk(n) =
d|n
μ(d)
log
n d
)k .
Theorem 2.48 (Selberg’s identity). We have Λk = Λk− 1 log +Λk− 1 ∗ Λ. Proof. We have Λk = μ ∗ logk^ = (μ ∗ logk−^1 ) log −μ log ∗ logk−^1 = Λk− 1 log +(−μ log) ∗ (μ ∗ u) ∗ logk−^1 = Λk− 1 log +Λ ∗ Λk− 1.
This completes the proof.
2.6. THE RIEMANN ZETA FUNCTION AND GENERATING FUNCTION 19
Corollary 2.51. We have ∑^ ∞
n=
μ(n) ns^
∏^ ∞
p=
ps
for Re(s) > 1.
Proof. This follows from Theorems 2.49 and 2.50.
Theorem 2.52 (generating function of the divisor function). We have
ζ^2 (s) =
∑^ ∞
n=
τ (n) ns^
for Re(s) > 1.
Proof. This follows from τ = u ∗ u.
Theorem 2.53 (generating function of the Euler totient function). We have
ζ(s − 1) ζ(s)
∑^ ∞
n=
ϕ(n) ns^
for Re(s) > 2.
Proof. This follows from ϕ = μ ∗ N.
Theorem 2.54 (generating function of the Mangoldt function). We have
ζ′(s) ζ(s)
∑^ ∞
n=
Λ(n) ns^
for Re(s) > 1.
Proof. By the Euler product formula, we have
log ζ(s) = log
∑^ ∞
n=
ns^
= log
p
ps
p
log
ps
Then by derivation on the both sides of the above equation, we get
ζ′(s) ζ(s)
p
1 − (^) p^1 s
)p−s^ log p
p
p−s^ log p
ps^
p^2 s^
p
∑^ ∞
k=
log p pks^
∑^ ∞
n=
Λ(n) ns^
Hence we have
−
ζ′(s) ζ(s)
∑^ ∞
n=
Λ(n) ns^
20 2. ARITHMETIC FUNCTIONS
2.7. Additive functions Definition 2.55. An arithmetic function f which is not identically zero is said to be additive if f (mn) = f (m) + f (n) (2.15)
whenever (m, n) = 1. Moreover, if (2.15) holds for all m, n, then f is called com- pletely additive.
Theorem 2.56. The function ω is additive. The function Ω is completely additive.
Proof. Write m = pa 1 1 · · · pa r r and n = qb 11 · · · q sbs with prime pi, qj and positive integers ai, bj. We have Ω(mn) =
i ai^ +^
j bj^ = Ω(m) + Ω(n). If (m, n) = 1, then ω(mn) = r + s = ω(m) + ω(n).
