This module discusses arithmetic functions and multiplicative functions. It defines an arithmetic function as a real or complex valued function on positive integers. A multiplicative function is an arithmetic function where the function value at a product of integers is equal to the product of the function values if the integers are relatively prime. It provides examples of common multiplicative functions including the Euler phi function, Möbius function, and divisor functions. The module aims to help students understand the definition of multiplicative functions and be able to determine if a given function is multiplicative or completely multiplicative.

1. Camarines Norte State College
Jose R. Abaño Campus
College of Education
Daet, Camarines Norte
Arithmetic Function: Multiplicative Function,
Definition and Basic Examples
Module13:
I. What this module is all about?
This module is all about Arithmetic Function: Multiplicative Function,
Definition and their Basic Examples. It is presented simply for the easy
understanding of the topic. Prior knowledge would be of great help in the process .
II. What are you expected to learn?
This module is designed to be able to:
1. Define arithmetic and multiplicative function.
2. Identify the basic important arithmetic function examples.
3. Differentiate multiplicative function from completely multiplicative.
4. Determine the important multiplicative function.
5. Solve basic multiplicative function problem.
III. Lesson 1: Arithmetic Function: Introduction and Basic Examples
A. Definition:
 An arithmetic function is any real- or complex-valued function
defined on the set N of positive integers.
 An arithmetic function is a function whose domain is defined on the
set of positive integers.
 It is just a sequence of real or complex numbers.
1 | Module 13: Arithmetic Function: Multiplicative Function, Definition and Basic Examples

2.  f: z + ⟶ c.
B. Basic Important Arithmetic Function Examples:
Function value at n
value at pm
properties
ω(n) (number
of distinct
prime
factors)
0 if n = 1,
k if Π 푃 푘푖
=1 iai
1
additive
Ω(n) (total
number of
prime
factors)
0 if n = 1,
Σ 푎 푘푖
=1 I if
n = Π 푃 푘푖
=1 i
m
completely
additive
log n
(logarithm)
log n
log pm
completely
additive
Λ(n) (von
Mangoldt
function)
log p if n = pm,
0 if n is not a
prime power
log p
neither
additive nor
multiplicative
log = Λ * 1
IV. Lesson 2: Multiplicative Function: Definitions and Properties
A. Definition:
 An arithmetic function f is called multiplicative if f is not congruent to 0.
 f is also multiplicative if f (ab) = f(a) f(b) for all a; b ⋲ N such that (a; b) =
1.
 An arithmetic function f is called completely multiplicative if f(ab) =
f(a)f(b) = 1.
 A multiplicative function is uniquely determined by its values on prime
powers, and a completely multiplicative function is uniquely
determined by its values on primes.
B. Properties:
a) Theorem 1.1 (Characterization of multiplicative functions). An
arithmetic function f is multiplicative if and only if f(1) = 1 and, for n ≥ 2.
f (n) = Π푝푚ll푛 푓(푝푚)
2 | Module 13: Arithmetic Function: Multiplicative Function, Definition and Basic Examples

3. b) Theorem 1.2 (Products and quotients of multiplicative functions).
Assume f and g is multiplicative function. Then:
i. The (point wise) product fg defined by (fg)(n) = f(n)g(n) is
multiplicative.
ii. If g is non-zero, then the quotient f=g (again defined point wise) is
multiplicative.
c) Some Important Multiplicative Function
Function
value at n
value at pm
properties
e(n)
1 if n = 1,
0 else
0
Unit element w.r.t.
Dirichlet product,
e * f = f *e = f
id(n)
(identity
function)
n
pm
s(n) (char. fct.
of squares)
1 if n = m2
with m ε N,
0 else
1 if m is even,
0 if m is odd
μ2(n) (char.
fct. of
squarefree
integers)
1 if n is
squarefree,
0 else
1 if m = 1,
0 if m > 1
μ(n) (Moebius
function)
1 if n = 1,
(-1)k if
n = Π 푃 푘푖
=1 i
(pi distinct)
0 otherwise
−1 if m = 1,
0 if m > 1
Σ푑l푛 휇푑 = 0 if
n ≥ 2
μ * 1 = e
λ(n) (Liouville
function)
1 if n = 1,
(-1) Σ 푎 푘푖
=1 I if
n = Π 푃 푘푖
=1 i
(−1)m
Σ푑l푛 휆푑 = s(n)
λ * 1 = s
φ(n) (Euler phi
function)
# ⦃1 ≤ m ≤ n :
(m, n) = 1⦄ pm(1 − 1/p)
Σ푑l푛 휑푑 = n
φ * 1 = id
d(n) (= τ (n))
(divisor
function)
Σ 1
푑l푛
m + 1 d = 1 * 1
σ(n) (sum of
divisor
function)
Σ 푑
푑l푛
푝푚+1 − 1
푝 − 1
σ = 1 * id
3 | Module 13: Arithmetic Function: Multiplicative Function, Definition and Basic Examples

4. V. Assessment:
A. Exercises: Tell whether the following is multiplicative, completely
multiplicative or not.
1. f (144)
2. f (5, 6)
3. f (0, 3)
4. f (25)
5. f (36)
6. f (0)
7. f (210)
8. f (9, 100)
9. f (n, 1), n ≠ 0
10. f (164)
B. Think and Apply:
1. Determine whether the arithmetic functions f(n) = n! and g(n) = n=2 are
completely multiplicative or not.
2. Define the arithmetic function g(n) by the following. g(n)=1 if n = 1 and 0
for n > 1. Prove that g(n) is multiplicative.
Prepared by:
Pelaosa, Larino Jr. Salazar
BSEd/Math 3A
4 | Module 13: Arithmetic Function: Multiplicative Function, Definition and Basic Examples