Elementary Number Theory
Franz Luef
Franz Luef
MA1301
Summary – Chapter 6
In this chapter we are discuss arithemtic functions.
Definition
A function f mapping a set of natural numbers to integers is
an arithmetic function.
An arithmetic function f is called multiplicative if
f (mn) = f (m)f (n) for gcd(m, n) = 1.
Examples
τ (n) denotes the number of divisors of a natural number n.
σ(n) denotes the sum of divisors of a natural number n.
Franz Luef
MA1301
Summary – Chapter 6
In this chapter we are discuss arithemtic functions.
Definition
A function f mapping a set of natural numbers to integers is
an arithmetic function.
An arithmetic function f is called multiplicative if
f (mn) = f (m)f (n) for gcd(m, n) = 1.
Examples
τ (n) denotes the number of divisors of a natural number n.
σ(n) denotes the sum of divisors of a natural number n.
Euler’s ϕ-function ϕ(n) denotes the number of integers a
relatively prime to n with 1 ≤ a ≤ n.
Franz Luef
MA1301
Summary – Chapter 6
In this chapter we are discuss arithemtic functions.
Definition
A function f mapping a set of natural numbers to integers is
an arithmetic function.
An arithmetic function f is called multiplicative if
f (mn) = f (m)f (n) for gcd(m, n) = 1.
Examples
τ (n) denotes the number of divisors of a natural number n.
σ(n) denotes the sum of divisors of a natural number n.
Euler’s ϕ-function ϕ(n) denotes the number of integers a
relatively prime to n with 1 ≤ a ≤ n.
Möbius function µ(n).
Franz Luef
MA1301
Summary – Chapter 6
In this chapter we are discuss arithemtic functions.
Definition
A function f mapping a set of natural numbers to integers is
an arithmetic function.
An arithmetic function f is called multiplicative if
f (mn) = f (m)f (n) for gcd(m, n) = 1.
Examples
τ (n) denotes the number of divisors of a natural number n.
σ(n) denotes the sum of divisors of a natural number n.
Euler’s ϕ-function ϕ(n) denotes the number of integers a
relatively prime to n with 1 ≤ a ≤ n.
Möbius function µ(n).
Franz Luef
MA1301
Summary – Chapter 6
The theory of multiplicative function relies largely on the prime
factorization of a natural number.
Lemma
The divisors of n = p1k1 · · · prkr are the numbers d = p1a1 · · · prar
for 0 ≤ ai ≤ ki for all i = 1, ..., r .
Theorem
For n = p1k1 · · · prkr
τ (n) = (k1 + 1) · · · (kr + 1)
Franz Luef
MA1301
Summary – Chapter 6
The theory of multiplicative function relies largely on the prime
factorization of a natural number.
Lemma
The divisors of n = p1k1 · · · prkr are the numbers d = p1a1 · · · prar
for 0 ≤ ai ≤ ki for all i = 1, ..., r .
Theorem
For n = p1k1 · · · prkr
τ (n) = (k1 + 1) · · · (kr + 1)
k +1
σ(n) =
p1 1 −1
p1 −1
kr +1
· · · prpr −1−1
Franz Luef
MA1301
Summary – Chapter 6
The theory of multiplicative function relies largely on the prime
factorization of a natural number.
Lemma
The divisors of n = p1k1 · · · prkr are the numbers d = p1a1 · · · prar
for 0 ≤ ai ≤ ki for all i = 1, ..., r .
Theorem
For n = p1k1 · · · prkr
τ (n) = (k1 + 1) · · · (kr + 1)
k +1
σ(n) =
p1 1 −1
p1 −1
kr +1
· · · prpr −1−1
τ and σ are multiplicative.
Franz Luef
MA1301
Summary – Chapter 6
The theory of multiplicative function relies largely on the prime
factorization of a natural number.
Lemma
The divisors of n = p1k1 · · · prkr are the numbers d = p1a1 · · · prar
for 0 ≤ ai ≤ ki for all i = 1, ..., r .
Theorem
For n = p1k1 · · · prkr
τ (n) = (k1 + 1) · · · (kr + 1)
k +1
σ(n) =
p1 1 −1
p1 −1
kr +1
· · · prpr −1−1
τ and σ are multiplicative.
τ and σ are determined by τ (p k ) and σ(p k ).
Franz Luef
MA1301
Summary – Chapter 6
The theory of multiplicative function relies largely on the prime
factorization of a natural number.
Lemma
The divisors of n = p1k1 · · · prkr are the numbers d = p1a1 · · · prar
for 0 ≤ ai ≤ ki for all i = 1, ..., r .
Theorem
For n = p1k1 · · · prkr
τ (n) = (k1 + 1) · · · (kr + 1)
k +1
σ(n) =
p1 1 −1
p1 −1
kr +1
· · · prpr −1−1
τ and σ are multiplicative.
τ and σ are determined by τ (p k ) and σ(p k ).
Franz Luef
MA1301
Summary – Chapter 6
Lemma
Suppose f is a multiplicative arithmetic function. Then
f (1) = 1 follows from f (1 · n) = f (1)f (n).
A multiplicative arithmetic function f is determined by
f (p k ).
Franz Luef
MA1301
Summary – Chapter 6
Lemma
Suppose f is a multiplicative arithmetic function. Then
f (1) = 1 follows from f (1 · n) = f (1)f (n).
A multiplicative arithmetic function f is determined by
f (p k ).
Suppose f and g are arithmetic multiplicative functions.
Then f · g is an arithmetic multiplicative function. In the
case that g (n) 6= 0 then f /g is an arithmetic
multiplicative function.
Franz Luef
MA1301
Summary – Chapter 6
Lemma
Suppose f is a multiplicative arithmetic function. Then
f (1) = 1 follows from f (1 · n) = f (1)f (n).
A multiplicative arithmetic function f is determined by
f (p k ).
Suppose f and g are arithmetic multiplicative functions.
Then f · g is an arithmetic multiplicative function. In the
case that g (n) 6= 0 then f /g is an arithmetic
multiplicative function.
Franz Luef
MA1301
Summary – Chapter 6
Möbius function
µ(1) = 1
µ(p1 · · · pr ) = (−1)r
µ(n) = 0 if p 2 |n for some prime number p.
Lemma
The Möbius function µ is multiplicative and it satisfies
X
µ(1) = 1 and
µ(d) = 0 for all n > 1.
d|n
Franz Luef
MA1301
Summary – Chapter 6
Möbius function
µ(1) = 1
µ(p1 · · · pr ) = (−1)r
µ(n) = 0 if p 2 |n for some prime number p.
Lemma
The Möbius function µ is multiplicative and it satisfies
X
µ(1) = 1 and
µ(d) = 0 for all n > 1.
d|n
Franz Luef
MA1301
Summary – Chapter 6
Sum function
P
F (n) = d|n f (d)
f is multiplicative if and only if the sum function F is
multiplicative.
Franz Luef
MA1301
Summary – Chapter 6
Sum function
P
F (n) = d|n f (d)
f is multiplicative if and only if the sum function F is
multiplicative.
For a multiplicative function f we have that
F (n) = Πri=1 (1 + f (pi ) + f (pi2 ) + · · · + f (prkr )).
Franz Luef
MA1301
Summary – Chapter 6
Sum function
P
F (n) = d|n f (d)
f is multiplicative if and only if the sum function F is
multiplicative.
For a multiplicative function f we have that
F (n) = Πri=1 (1 + f (pi ) + f (pi2 ) + · · · + f (prkr )).
Möbius Inversion Formula
P
Suppose F (n) = d|n f (d). Then
f (n) =
X
d|n
Franz Luef
n
µ( )F (d).
d
MA1301
Summary – Chapter 6
Sum function
P
F (n) = d|n f (d)
f is multiplicative if and only if the sum function F is
multiplicative.
For a multiplicative function f we have that
F (n) = Πri=1 (1 + f (pi ) + f (pi2 ) + · · · + f (prkr )).
Möbius Inversion Formula
P
Suppose F (n) = d|n f (d). Then
f (n) =
X
d|n
Franz Luef
n
µ( )F (d).
d
MA1301