Arithmetic and Multiplicative Function

1. Prayer

2. Arithmetic and
Multiplicative Function
Math 207
Number Theory
Presented by: Allanah Jane D. Bay-ongan

3. At the end of the lesson learners will be
able to:
 define Arithmetic and multiplicative function
 differentiate multiplicative function and
completely multiplicative function.
 solve problems involving multiplicative
functions.
 solve problems involving Tau, Sigma and Mobius
Function as multiplicative function.

4. Energizer

5. Energizer

19. Tt

20. Tt

27. Amazing!

28. Arithmetic Function
a function of the form f : →
•Arithmetic function maps positive integers to complex numbers, crucial in
number theory
•Domain consists of positive integers (natural numbers), forming the input set
•Codomain encompasses complex numbers, representing possible output values
•Bounded function maintains outputs within a fixed range, never exceeding
certain limits

29. Few well known Arithmetic Function
U(n)=1
• For all N
N(n)=
n
• For all N
I(n)=
• If n

30. Few well known Arithmetic Function
Constant
Function
U(n)=1
• For all N
N(n)=
n
• For all
N
I(n)=
• If n

31. Few well known Arithmetic Function
U(n)=1
• For all
N
Identity
Function
N(n)=n
• For all
N
I(n)=
• If n

32. Few well known Arithmetic Function
U(n)=1
• For all
N
N(n)=n
• For all
N
Indicator
Function
I(n)=
• If n

33. Few well known Arithmetic Function
𝜏(n)
• the
number of
all divisor
of n
(d(n)),
where n ∈
Z, n 0.
(
σ n)
• the sum
of all
divisor of
n, where
n ∈ Z, n
0.
μ(n)
• the sum
of the
primitive
nth roots
of unity

34. Few well known Arithmetic Function
𝜏(n)
• the number of all
divisor of n (d(n)),
where n ∈ Z, n 0.
(
σ n)
• the sum of
all divisor
of n,
where n ∈
Z, n 0.
μ(n)
• the sum of
the
primitive
nth roots
of unity

35. TAU n
(n)
𝜏
- Number of positive divisor of n,
(10)
τ
(divisors: 1,2,5,10)
=
=1+1+1+1
4
(
τ 3)
(divisors: 1,3)
=
=1+1
2

36. TAU n
𝜏(n)
- Number of positive divisor of n,
(2)=2
τ
(divisors: 1, 2)
(
τ 16)=5
(divisors: 1, 2,4,8,16)
n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
τn 1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5
𝜏(n)=2

37. Formula for (
τ n)
If
=
(2+1)(2+1)(1+1)
=3*3*2
=18
=
(1+1)(1+1)(1+1)
=2*2*2
=8

38. Few well known Arithmetic Function
𝜏(n)
• the
number of
all divisor
of n
(d(n)),
where n ∈
Z, n 0.
(
σ n)
• the sum
of all
divisor
of n,
where n
∈ Z, n
0.
μ(n)
• the sum
of the
primitiv
e nth ro
ots of
unity

39. Few well known Arithmetic Function
𝜏(n
)
• the number of
all divisor of n
(d(n)), where
n ∈ Z, n 0.
(
σ n)
• the sum of all divisor
of n, where n ∈ Z, n
0.
μ(n
)
• the sum of
the
primitive nth
roots of
unity

40. SIGMA n
(
σ n)
- Sum of the divisor of n,
σ(9)=
(divisors : 1, 3, 9)
= 1+ 3+9
σ(9)=13
σ(12)=
(divisors: 1, 2,3,4,12)
= 1+ 2+3+4+12
= 28
n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
σn 1 3 4 7 6 12 8 15 13 18 13 28 15 24 24 31

41. σ(180)
=
=
=7*13*6
=546

42. σ(4,328)
=
=
=8,130

43. σ(18,909)
= 11 191
⋅ ⋅
=
=192
=29,952

44. Few well known Arithmetic Function
𝜏(n)
• the
number of
all divisor
of n (d(n)),
where n ∈
Z, n 0.
(
σ n)
• the sum
of all
divisor of
n, where
n ∈ Z, n
0.
μ(n)
• the sum
of the
primitive
nth roots
of unity

45. Few well known Arithmetic Function
𝜏(n)
• the number of all
divisor of n (d(n)),
where n ∈ Z, n 0.
(
σ n)
• the sum of all
divisor of n, where
n ∈ Z, n 0.
μ(n)
• the sum of the
primitive nth roots
of unity

46. Möbius Function
μ(n)
- the sum of the primitive nth roots of unity
- (1)
μ is defined to be 1.
- (n) = 1
μ if n is a square-free positive integer with an even number of prime factors.
- (n) = −1
μ if n is a square-free positive integer with an odd number of prime factors.
- (n) = 0
μ if n has a squared prime factor.
August Ferdinand
Möbius

47. n 1 2 3 4 5 6 7 8 9 10
μn 1 -1 -1 0 -1 1 -1 0 0 1
The Zero Bin
The “Zero” bin has
multiples of square numbers
(excluding 1):
{4, 8, 9, 12, 16, 18, 20, 24,…}.
These can be written in terms of μ:
•μ(4) = 0,
•μ(8) = 0,
•μ(9) = 0,
•μ(12) = 0
The -1 Bin
This bin contains every prime number. It
also contains any number that factors into
an odd number of distinct primes; In other
words, it’s a number where 3, 5, 7,…
primes are multiplied together. For
example:
•μ (30)
•μ (66 )
•μ (715 )
•μ (2730)
The +1 Bin
This bin has any number
that factors into an even
number of distinct primes.
For example:
•μ (6)
•μ (22)
•μ (65)
•μ (210)

48. n 1 2 3 4 5 6 7 8 9 10
μn 1 -1 -1 0 -1 1 -1 0 0 1
The Zero Bin
The “Zero” bin has
multiples of square numbers (excluding
1):
{4, 8, 9, 12, 16, 18, 20, 24,…}.
These can be written in terms of μ:
•μ(4) = 0,
•μ(8) = 0,
•μ(9) = 0,
•μ(12) = 0
The +1 Bin
This bin has
any number
that factors
into an even
number of
distinct
primes. For
example:
•6 (2 * 3 )
•22 (2 * 11)
•65 (5 * 13)
•210 (2 * 3 *
5 * 7)
The -1 Bin
This bin contains every
prime number. It also
contains any number
that factors into an odd
number of distinct
primes; In other words,
it’s a number where 3,
5, 7,… primes are
multiplied together. For
example:
•30 (2 * 3 * 5)
•66 (2 * 3 * 11)
•715 (5 * 11 * 13)
•2730 (2 * 3 * 5 * 7 *
13)

49. n 1 2 3 4 5 6 7 8 9 10
μn 1 -1 -1 0 -1 1 -1 0 0 1
The Zero Bin
The “Zero” bin
has
multiples of s
quare numbe
rs
(excluding 1):
{4, 8, 9, 12,
16, 18, 20, 24,
…}.
These can be
written in
terms of μ:
•μ(4) = 0,
•μ(8) = 0,
•μ(9) = 0,
•μ(12) = 0
The +1 Bin
This bin has
any number
that factors
into an even
number of
distinct
primes. For
example:
•6 (2 * 3 )
•22 (2 * 11)
•65 (5 * 13)
•210 (2 * 3 *
5 * 7)
The -1 Bin
This bin contains every prime number. It also contains
any number that factors into an odd number of distinct
primes; In other words, it’s a number where 3, 5, 7,…
primes are multiplied together. For example:
•30 (2 * 3 * 5)
•66 (2 * 3 * 11)
•715 (5 * 11 * 13)
•2730 (2 * 3 * 5 * 7 * 13)

50. n 1 2 3 4 5 6 7 8 9 10
μn 1 -1 -1 0 -1 1 -1 0 0 1
The Zero Bin
The “Zero” bin has
multiples of squar
e numbers
(excluding 1):
{4, 8, 9, 12, 16, 18,
20, 24,…}.
These can be
written in terms of μ:
•μ(4) = 0,
•μ(8) = 0,
•μ(9) = 0,
•μ(12) = 0
The +1 Bin
This bin has any number that
factors into an even number of
distinct primes. For example:
•6 (2 * 3 )
•22 (2 * 11)
•65 (5 * 13)
•210 (2 * 3 * 5 * 7)
The -1 Bin
This bin contains every
prime number. It also
contains any number
that factors into an odd
number of distinct
primes; In other words,
it’s a number where 3, 5,
7,… primes are
multiplied together. For
example:
•30 (2 * 3 * 5)
•66 (2 * 3 * 11)
•715 (5 * 11 * 13)
•2730 (2 * 3 * 5 * 7 * 13)

51. Let’s try this
•μ(300)
•μ(617)
•μ(8910)
•μ(210)
=
=prime number
=
=2*3*5*7
•μ(300) = 0
•μ(617) = -1
•μ(8910) = 0
•μ(210)= 1

52. Multiplicative Function
An arithmetic function is called multiplicative if f is not
congruent to 0.
 f is also multiplicative if f(mn) = f(m) f(n) for all m; n ∈
N such that (m;n) = 1
 Whenever gcd (m,n) =1, when they are coprime.
two or more integers that
have no common factors
other than 1

53.  A multiplicative function is completely determined by
its behavior on the prime powers.
An arithmetic function is called completely
multiplicative if f(1) = 1 and f(mn) = f(m) f(n), holds for
all positive integers m and n, even when they are not
coprime. ( f(n)=n, f(n)=na
)

54. Multiplicative Function
Given: f(15)
f(mn) = f(m) f(n)
f(15) = f(5) f(3)
f(m) = 5
f(n) = 3
gcd(m,n) = 1
Completely Multiplicative
Function
Given: f(164)
f(mn) = f(m) f(n)
f(164) = f(22
) f(41)
f(m) = 22
= 4
f(n) = 41
gcd(m,n) = 1
( f(n)=n, f(n)=na
)
f(164) = f(4) f(41) is completely
multiplicative function.

55. Multiplicative Function
Given: f(6)
f(mn) = f(m) f(n)
f(6) = f(3) f(2)
f(m) = 3
f(n) = 2
gcd(m,n) = 1
Completely Multiplicative
Function
Given: f(25)
f(mn) = f(m) f(n)
f(25) = f(52
)
f(m) = 5
f(n) = 5
gcd(m,n) = 5
f(25) = f(5) f(5) is completely
multiplicative function.

56. Tau( ) , Sigma (
𝜏 ) and
σ
Mobius Function (μ) as
Multiplicative Function

57. 𝑛=𝑝 1𝑘 1
… 𝑝𝑟 𝑘𝑟
= =f(n)f(1)
f(1)= 1
f(1) is not identically zero
An Arithmetic function is multiplicative if f(mn) = f(m) f(n) ;
gcd(m,n) = 1

58. Theorem: The function and
𝜏 σ are both multiplicative
function
• Proof : to show
𝜏(mn) = 𝜏(m) 𝜏(n) ; gcd(m,n) = 1.
σ(mn) = σ(m) σ(n) ; gcd(m,n) = 1.
• )
=…
= (m) (n)
𝜏 𝜏
Hence; is a
𝜏 multiplicative function

59. Theorem: The function and
𝜏 σ are both multiplicative
function
• Proof : to show
𝜏(mn) = 𝜏(m) 𝜏(n) ; gcd(m,n) = 1.
σ(mn) = σ(m) σ(n) ; gcd(m,n) = 1.
• )
•
= σ(m) σ(n)
Hence; 𝜏 and σ are both multiplicative function

60. 3 and 4 are co-prime
(
τ 12)=6 (divisors: 1, 2, 3, 4,6,12)
𝜏(mn) = 𝜏(m) 𝜏(n) gcd(m,n) = 1.
𝜏(12) = 𝜏(3) 𝜏(4)
𝜏(12)=2 x 3
𝜏(12) = 6
𝜏(12) = 𝜏(2) 𝜏(6)
𝜏(12)=2 x 4
𝜏(12) = 8
2 and 6 are not co-prime
∴ 𝜏 (12) is multiplicative .
Example:

61. 4 and 3 are co-prime
(12)=6 (divisors: 1, 2, 3, 4,6,12)
(mn) = (m) (n) gcd(m,n) = 1.
(12) = (3) (4)
(12)=(1+3) (1+2+4)
(12)=(4) (7)
(12) = 28
(12) = (2) (6)
(12)=(1+2)(1+2+3+6)
(12)=(3)(12)
(12) = 36
2 and 6 are not co-prime
∴ σ (12 ) is multiplicative .
Example:

62. Theorem: The Mobius function as
Multiplicative Function
• Proof : to show μ(mn) = μ(m) μ(n) provided that the gcd(m,n) = 1.
Let
•
=
=
= (m) (n)
Hence; multiplicative function
Note: m=1, n=1
μ(mn) = μ(m) μ(n)
μ(1) = μ(1) μ(1)
1=1
μ(m) =0
μ(n) =0

63. Theorem: The Mobius function as
Multiplicative Function
• (3,4)
•
• = (-1) (0)
• = 0
Hence; multiplicative function
n 1 2 3 4 5 6 7 8 9 10
μn 1 -1 -1 0 -1 1 -1 0 0 1
• (6,2)
•
• = (1) (-1)
• = -1
3 and 4 are co-prime 2 and 6 are not co-prime

64. Another example of Multiplicative Function

65.  Multiplicative functions serve as fundamental tools in number
theory, particularly in analytic number theory, where they are
used to study properties of integers and their relationships.
 It reveal the structure of integers through prime factorization and
provide powerful tools for studying various number-theoretic
problems.
 The multiplicative property significantly simplifies calculations by
reducing the need to compute the function's value for large
integers. Instead, we can compute it for prime powers and then
combine these values to obtain the function's value for any
integer.

66. THANK YOU FOR
LISTENING!!!