Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Multiplicative Functions
Bernd Schröder
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Introduction
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Introduction
Definition.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Introduction
Definition. An arithmetic function is a function that is defined
for all positive integers.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Introduction
Definition. An arithmetic function is a function that is defined
for all positive integers.
Definition.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Introduction
Definition. An arithmetic function is a function that is defined
for all positive integers.
Definition. An arithmetic function is called multiplicative iff
(m, n) = 1 implies f (mn) = f (m)f (n).
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Introduction
Definition. An arithmetic function is a function that is defined
for all positive integers.
Definition. An arithmetic function is called multiplicative iff
(m, n) = 1 implies f (mn) = f (m)f (n). It is called completely
multiplicative iff for all m and n we have f (mn) = f (m)f (n).
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Introduction
Definition. An arithmetic function is a function that is defined
for all positive integers.
Definition. An arithmetic function is called multiplicative iff
(m, n) = 1 implies f (mn) = f (m)f (n). It is called completely
multiplicative iff for all m and n we have f (mn) = f (m)f (n).
In this presentation, we will see that Euler’s ϕ-function is
multiplicative
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Introduction
Definition. An arithmetic function is a function that is defined
for all positive integers.
Definition. An arithmetic function is called multiplicative iff
(m, n) = 1 implies f (mn) = f (m)f (n). It is called completely
multiplicative iff for all m and n we have f (mn) = f (m)f (n).
In this presentation, we will see that Euler’s ϕ-function is
multiplicative, we will consider the sum and number of divisor
functions
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Introduction
Definition. An arithmetic function is a function that is defined
for all positive integers.
Definition. An arithmetic function is called multiplicative iff
(m, n) = 1 implies f (mn) = f (m)f (n). It is called completely
multiplicative iff for all m and n we have f (mn) = f (m)f (n).
In this presentation, we will see that Euler’s ϕ-function is
multiplicative, we will consider the sum and number of divisor
functions, which are multiplicative
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Introduction
Definition. An arithmetic function is a function that is defined
for all positive integers.
Definition. An arithmetic function is called multiplicative iff
(m, n) = 1 implies f (mn) = f (m)f (n). It is called completely
multiplicative iff for all m and n we have f (mn) = f (m)f (n).
In this presentation, we will see that Euler’s ϕ-function is
multiplicative, we will consider the sum and number of divisor
functions, which are multiplicative, and we will explore the
relationship between a multiplicative function f and its
summary function F(n) := ∑ f (d).
d|n
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
s
Möbius Inversion
a
Theorem. If f is multiplicative and n = ∏ pj j is the prime
j=1
s
a
factorization of n, then f (n) = ∏ f pj j .
j=1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
s
Möbius Inversion
a
Theorem. If f is multiplicative and n = ∏ pj j is the prime
j=1
s
a
factorization of n, then f (n) = ∏ f pj j .
j=1
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
s
Möbius Inversion
a
Theorem. If f is multiplicative and n = ∏ pj j is the prime
j=1
s
a
factorization of n, then f (n) = ∏ f pj j .
j=1
Proof. Induction on s.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
s
Möbius Inversion
a
Theorem. If f is multiplicative and n = ∏ pj j is the prime
j=1
s
a
factorization of n, then f (n) = ∏ f pj j .
j=1
Proof. Induction on s. The base step for s = 1 is trivial.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
s
Möbius Inversion
a
Theorem. If f is multiplicative and n = ∏ pj j is the prime
j=1
s
a
factorization of n, then f (n) = ∏ f pj j .
j=1
Proof. Induction on s. The base step for s = 1 is trivial.
Induction step s → s + 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
s
Möbius Inversion
a
Theorem. If f is multiplicative and n = ∏ pj j is the prime
j=1
s
a
factorization of n, then f (n) = ∏ f pj j .
j=1
Proof. Induction on s. The base step for s = 1 is trivial.
s+1
a
Induction step s → s + 1. Let n = ∏ pj j .
j=1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
s
Möbius Inversion
a
Theorem. If f is multiplicative and n = ∏ pj j is the prime
j=1
s
a
factorization of n, then f (n) = ∏ f pj j .
j=1
Proof. Induction on s. The base step for s = 1 is trivial.
s+1
a
Induction step s → s + 1. Let n = ∏ pj j . Then
j=1
s+1
f (n) = f
!
a
∏ pj j
j=1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
s
Möbius Inversion
a
Theorem. If f is multiplicative and n = ∏ pj j is the prime
j=1
s
a
factorization of n, then f (n) = ∏ f pj j .
j=1
Proof. Induction on s. The base step for s = 1 is trivial.
s+1
a
Induction step s → s + 1. Let n = ∏ pj j . Then
j=1
s+1
f (n) = f
Multiplicative Functions
a
∏ pj j
j=1
Bernd Schröder
!
s
=f
!
a
∏ pj j
!
a
s+1
ps+1
j=1
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
s
Möbius Inversion
a
Theorem. If f is multiplicative and n = ∏ pj j is the prime
j=1
s
a
factorization of n, then f (n) = ∏ f pj j .
j=1
Proof. Induction on s. The base step for s = 1 is trivial.
s+1
a
Induction step s → s + 1. Let n = ∏ pj j . Then
j=1
s+1
f (n) = f
Multiplicative Functions
a
∏ pj j
j=1
Bernd Schröder
!
s
=f
!
a
∏ pj j
j=1
!
a
s+1
ps+1
s
=f
!
a
∏ pj j
a
s+1
f ps+1
j=1
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
s
Möbius Inversion
a
Theorem. If f is multiplicative and n = ∏ pj j is the prime
j=1
s
a
factorization of n, then f (n) = ∏ f pj j .
j=1
Proof. Induction on s. The base step for s = 1 is trivial.
s+1
a
Induction step s → s + 1. Let n = ∏ pj j . Then
j=1
s+1
f (n) = f
!
a
∏ pj j
s
j=1
s
=
a
f
∏ pj j
a
∏ pj j
=f
j=1
!
!
a
s+1
f ps+1
!
a
s+1
ps+1
s
=f
!
a
∏ pj j
a
s+1
f ps+1
j=1
j=1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
s
Möbius Inversion
a
Theorem. If f is multiplicative and n = ∏ pj j is the prime
j=1
s
a
factorization of n, then f (n) = ∏ f pj j .
j=1
Proof. Induction on s. The base step for s = 1 is trivial.
s+1
a
Induction step s → s + 1. Let n = ∏ pj j . Then
j=1
s+1
f (n) = f
!
a
∏ pj j
s
=f
j=1
s
=
a
f
∏ pj j
j=1
Bernd Schröder
Multiplicative Functions
!
a
∏ pj j
!
a
s+1
ps+1
s
=f
j=1
!
!
a
∏ pj j
a
s+1
f ps+1
j=1
s+1 a
as+1 = ∏ f pj j
f ps+1
j=1
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
s
Möbius Inversion
a
Theorem. If f is multiplicative and n = ∏ pj j is the prime
j=1
s
a
factorization of n, then f (n) = ∏ f pj j .
j=1
Proof. Induction on s. The base step for s = 1 is trivial.
s+1
a
Induction step s → s + 1. Let n = ∏ pj j . Then
j=1
s+1
f (n) = f
!
a
∏ pj j
s
=f
j=1
s
=
a
f
∏ pj j
j=1
Bernd Schröder
Multiplicative Functions
!
a
∏ pj j
!
a
s+1
ps+1
s
=f
j=1
!
!
a
∏ pj j
a
s+1
f ps+1
j=1
s+1 a
as+1 = ∏ f pj j
f ps+1
j=1
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. A positive integer is prime iff ϕ(p) = p − 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. A positive integer is prime iff ϕ(p) = p − 1.
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. A positive integer is prime iff ϕ(p) = p − 1.
Proof. “⇒.”
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. A positive integer is prime iff ϕ(p) = p − 1.
Proof. “⇒.” If p is prime, then for all a ∈ {1, . . . , p − 1} we
have (a, p) = 1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. A positive integer is prime iff ϕ(p) = p − 1.
Proof. “⇒.” If p is prime, then for all a ∈ {1, . . . , p − 1} we
have (a, p) = 1, so ϕ(p) = p − 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. A positive integer is prime iff ϕ(p) = p − 1.
Proof. “⇒.” If p is prime, then for all a ∈ {1, . . . , p − 1} we
have (a, p) = 1, so ϕ(p) = p − 1.
“⇐.”
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. A positive integer is prime iff ϕ(p) = p − 1.
Proof. “⇒.” If p is prime, then for all a ∈ {1, . . . , p − 1} we
have (a, p) = 1, so ϕ(p) = p − 1.
“⇐.” If ϕ(p) = p − 1, then (a, p) = 1 for all integers
a ∈ {1, . . . , p − 1}.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. A positive integer is prime iff ϕ(p) = p − 1.
Proof. “⇒.” If p is prime, then for all a ∈ {1, . . . , p − 1} we
have (a, p) = 1, so ϕ(p) = p − 1.
“⇐.” If ϕ(p) = p − 1, then (a, p) = 1 for all integers
a ∈ {1, . . . , p − 1}. But that means that p is prime.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. A positive integer is prime iff ϕ(p) = p − 1.
Proof. “⇒.” If p is prime, then for all a ∈ {1, . . . , p − 1} we
have (a, p) = 1, so ϕ(p) = p − 1.
“⇐.” If ϕ(p) = p − 1, then (a, p) = 1 for all integers
a ∈ {1, . . . , p − 1}. But that means that p is prime.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. A positive integer is prime iff ϕ(p) = p − 1.
Proof. “⇒.” If p is prime, then for all a ∈ {1, . . . , p − 1} we
have (a, p) = 1, so ϕ(p) = p − 1.
“⇐.” If ϕ(p) = p − 1, then (a, p) = 1 for all integers
a ∈ {1, . . . , p − 1}. But that means that p is prime.
Theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. A positive integer is prime iff ϕ(p) = p − 1.
Proof. “⇒.” If p is prime, then for all a ∈ {1, . . . , p − 1} we
have (a, p) = 1, so ϕ(p) = p − 1.
“⇐.” If ϕ(p) = p − 1, then (a, p) = 1 for all integers
a ∈ {1, . . . , p − 1}. But that means that p is prime.
Theorem. Let p be prime and let a be a positive integer. Then
ϕ (pa ) = pa − pa−1 .
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. A positive integer is prime iff ϕ(p) = p − 1.
Proof. “⇒.” If p is prime, then for all a ∈ {1, . . . , p − 1} we
have (a, p) = 1, so ϕ(p) = p − 1.
“⇐.” If ϕ(p) = p − 1, then (a, p) = 1 for all integers
a ∈ {1, . . . , p − 1}. But that means that p is prime.
Theorem. Let p be prime and let a be a positive integer. Then
ϕ (pa ) = pa − pa−1 .
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. A positive integer is prime iff ϕ(p) = p − 1.
Proof. “⇒.” If p is prime, then for all a ∈ {1, . . . , p − 1} we
have (a, p) = 1, so ϕ(p) = p − 1.
“⇐.” If ϕ(p) = p − 1, then (a, p) = 1 for all integers
a ∈ {1, . . . , p − 1}. But that means that p is prime.
Theorem. Let p be prime and let a be a positive integer. Then
ϕ (pa ) = pa − pa−1 .
Proof. A number x ∈ {1, . . . , pa } is so that (x, pa ) = 1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. A positive integer is prime iff ϕ(p) = p − 1.
Proof. “⇒.” If p is prime, then for all a ∈ {1, . . . , p − 1} we
have (a, p) = 1, so ϕ(p) = p − 1.
“⇐.” If ϕ(p) = p − 1, then (a, p) = 1 for all integers
a ∈ {1, . . . , p − 1}. But that means that p is prime.
Theorem. Let p be prime and let a be a positive integer. Then
ϕ (pa ) = pa − pa−1 .
Proof. A number x ∈ {1, . . . , pa } is so that (x, pa ) = 1 iff p is
not a prime factor of x.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. A positive integer is prime iff ϕ(p) = p − 1.
Proof. “⇒.” If p is prime, then for all a ∈ {1, . . . , p − 1} we
have (a, p) = 1, so ϕ(p) = p − 1.
“⇐.” If ϕ(p) = p − 1, then (a, p) = 1 for all integers
a ∈ {1, . . . , p − 1}. But that means that p is prime.
Theorem. Let p be prime and let a be a positive integer. Then
ϕ (pa ) = pa − pa−1 .
Proof. A number x ∈ {1, . . . , pa } is so that (x, pa ) = 1 iff p is
a
not a prime factor of x. There are pp = pa−1 multiples of p in
the set {1, . . . , pa }.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. A positive integer is prime iff ϕ(p) = p − 1.
Proof. “⇒.” If p is prime, then for all a ∈ {1, . . . , p − 1} we
have (a, p) = 1, so ϕ(p) = p − 1.
“⇐.” If ϕ(p) = p − 1, then (a, p) = 1 for all integers
a ∈ {1, . . . , p − 1}. But that means that p is prime.
Theorem. Let p be prime and let a be a positive integer. Then
ϕ (pa ) = pa − pa−1 .
Proof. A number x ∈ {1, . . . , pa } is so that (x, pa ) = 1 iff p is
a
not a prime factor of x. There are pp = pa−1 multiples of p in
the set {1, . . . , pa }.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If (m, n) = 1, then ϕ(mn) = ϕ(m)ϕ(n).
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If (m, n) = 1, then ϕ(mn) = ϕ(m)ϕ(n).
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If (m, n) = 1, then ϕ(mn) = ϕ(m)ϕ(n).
Proof. First note that
[
{1, . . . , mn} =
{dm + r : d ∈ {0, . . . , n − 1}}.
r∈{1,...,m}
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If (m, n) = 1, then ϕ(mn) = ϕ(m)ϕ(n).
Proof. First note that
[
{1, . . . , mn} =
{dm + r : d ∈ {0, . . . , n − 1}}. If
r∈{1,...,m}
(m, r) = b 6= 1, then, for all x ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have
(x, mn)
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If (m, n) = 1, then ϕ(mn) = ϕ(m)ϕ(n).
Proof. First note that
[
{1, . . . , mn} =
{dm + r : d ∈ {0, . . . , n − 1}}. If
r∈{1,...,m}
(m, r) = b 6= 1, then, for all x ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have
(x, mn) ≥ b
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If (m, n) = 1, then ϕ(mn) = ϕ(m)ϕ(n).
Proof. First note that
[
{1, . . . , mn} =
{dm + r : d ∈ {0, . . . , n − 1}}. If
r∈{1,...,m}
(m, r) = b 6= 1, then, for all x ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have
(x, mn) ≥ b > 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If (m, n) = 1, then ϕ(mn) = ϕ(m)ϕ(n).
Proof. First note that
[
{1, . . . , mn} =
{dm + r : d ∈ {0, . . . , n − 1}}. If
r∈{1,...,m}
(m, r) = b 6= 1, then, for all x ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have
(x, mn) ≥ b > 1. If (m, r) = 1, then, for all
y ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have (y, m) = 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If (m, n) = 1, then ϕ(mn) = ϕ(m)ϕ(n).
Proof. First note that
[
{1, . . . , mn} =
{dm + r : d ∈ {0, . . . , n − 1}}. If
r∈{1,...,m}
(m, r) = b 6= 1, then, for all x ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have
(x, mn) ≥ b > 1. If (m, r) = 1, then, for all
y ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have (y, m) = 1. Now recall that,
for r fixed, because (m, n) = 1, {dm + r : d ∈ {0, . . . , n − 1}} is a
complete set of nonnegative residues modulo n.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If (m, n) = 1, then ϕ(mn) = ϕ(m)ϕ(n).
Proof. First note that
[
{1, . . . , mn} =
{dm + r : d ∈ {0, . . . , n − 1}}. If
r∈{1,...,m}
(m, r) = b 6= 1, then, for all x ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have
(x, mn) ≥ b > 1. If (m, r) = 1, then, for all
y ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have (y, m) = 1. Now recall that,
for r fixed, because (m, n) = 1, {dm + r : d ∈ {0, . . . , n − 1}} is a
complete set of nonnegative residues modulo n. Therefore, exactly
ϕ(n) elements y ∈ {dm + r : d ∈ {0, . . . , n − 1}} satisfy (y, n) = 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If (m, n) = 1, then ϕ(mn) = ϕ(m)ϕ(n).
Proof. First note that
[
{1, . . . , mn} =
{dm + r : d ∈ {0, . . . , n − 1}}. If
r∈{1,...,m}
(m, r) = b 6= 1, then, for all x ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have
(x, mn) ≥ b > 1. If (m, r) = 1, then, for all
y ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have (y, m) = 1. Now recall that,
for r fixed, because (m, n) = 1, {dm + r : d ∈ {0, . . . , n − 1}} is a
complete set of nonnegative residues modulo n. Therefore, exactly
ϕ(n) elements y ∈ {dm + r : d ∈ {0, . . . , n − 1}} satisfy (y, n) = 1.
Because (y, n) = 1 and (y, m) = 1, every one of these elements
satisfies (y, mn) = 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If (m, n) = 1, then ϕ(mn) = ϕ(m)ϕ(n).
Proof. First note that
[
{1, . . . , mn} =
{dm + r : d ∈ {0, . . . , n − 1}}. If
r∈{1,...,m}
(m, r) = b 6= 1, then, for all x ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have
(x, mn) ≥ b > 1. If (m, r) = 1, then, for all
y ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have (y, m) = 1. Now recall that,
for r fixed, because (m, n) = 1, {dm + r : d ∈ {0, . . . , n − 1}} is a
complete set of nonnegative residues modulo n. Therefore, exactly
ϕ(n) elements y ∈ {dm + r : d ∈ {0, . . . , n − 1}} satisfy (y, n) = 1.
Because (y, n) = 1 and (y, m) = 1, every one of these elements
satisfies (y, mn) = 1.
In summary, there are ϕ(m) sets among the sets
{dm + r : d ∈ {0, . . . , n − 1}} that contain elements whose greatest
common divisor with mn could be 1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If (m, n) = 1, then ϕ(mn) = ϕ(m)ϕ(n).
Proof. First note that
[
{1, . . . , mn} =
{dm + r : d ∈ {0, . . . , n − 1}}. If
r∈{1,...,m}
(m, r) = b 6= 1, then, for all x ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have
(x, mn) ≥ b > 1. If (m, r) = 1, then, for all
y ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have (y, m) = 1. Now recall that,
for r fixed, because (m, n) = 1, {dm + r : d ∈ {0, . . . , n − 1}} is a
complete set of nonnegative residues modulo n. Therefore, exactly
ϕ(n) elements y ∈ {dm + r : d ∈ {0, . . . , n − 1}} satisfy (y, n) = 1.
Because (y, n) = 1 and (y, m) = 1, every one of these elements
satisfies (y, mn) = 1.
In summary, there are ϕ(m) sets among the sets
{dm + r : d ∈ {0, . . . , n − 1}} that contain elements whose greatest
common divisor with mn could be 1, and each of these sets contains
exactly ϕ(n) such elements.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If (m, n) = 1, then ϕ(mn) = ϕ(m)ϕ(n).
Proof. First note that
[
{1, . . . , mn} =
{dm + r : d ∈ {0, . . . , n − 1}}. If
r∈{1,...,m}
(m, r) = b 6= 1, then, for all x ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have
(x, mn) ≥ b > 1. If (m, r) = 1, then, for all
y ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have (y, m) = 1. Now recall that,
for r fixed, because (m, n) = 1, {dm + r : d ∈ {0, . . . , n − 1}} is a
complete set of nonnegative residues modulo n. Therefore, exactly
ϕ(n) elements y ∈ {dm + r : d ∈ {0, . . . , n − 1}} satisfy (y, n) = 1.
Because (y, n) = 1 and (y, m) = 1, every one of these elements
satisfies (y, mn) = 1.
In summary, there are ϕ(m) sets among the sets
{dm + r : d ∈ {0, . . . , n − 1}} that contain elements whose greatest
common divisor with mn could be 1, and each of these sets contains
exactly ϕ(n) such elements. Thus ϕ(mn) = ϕ(m)ϕ(n).
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If (m, n) = 1, then ϕ(mn) = ϕ(m)ϕ(n).
Proof. First note that
[
{1, . . . , mn} =
{dm + r : d ∈ {0, . . . , n − 1}}. If
r∈{1,...,m}
(m, r) = b 6= 1, then, for all x ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have
(x, mn) ≥ b > 1. If (m, r) = 1, then, for all
y ∈ {dm + r : d ∈ {0, . . . , n − 1}}, we have (y, m) = 1. Now recall that,
for r fixed, because (m, n) = 1, {dm + r : d ∈ {0, . . . , n − 1}} is a
complete set of nonnegative residues modulo n. Therefore, exactly
ϕ(n) elements y ∈ {dm + r : d ∈ {0, . . . , n − 1}} satisfy (y, n) = 1.
Because (y, n) = 1 and (y, m) = 1, every one of these elements
satisfies (y, mn) = 1.
In summary, there are ϕ(m) sets among the sets
{dm + r : d ∈ {0, . . . , n − 1}} that contain elements whose greatest
common divisor with mn could be 1, and each of these sets contains
exactly ϕ(n) such elements. Thus ϕ(mn) = ϕ(m)ϕ(n).
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
j=1
s s
1
a −1
ϕ(n) = n ∏ 1 −
= ∏ pj j (pj − 1)
pj
j=1
j=1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
j=1
s s
1
a −1
ϕ(n) = n ∏ 1 −
= ∏ pj j (pj − 1)
pj
j=1
j=1
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
j=1
s s
1
a −1
ϕ(n) = n ∏ 1 −
= ∏ pj j (pj − 1)
pj
j=1
j=1
Proof. Induction on s.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
j=1
s s
1
a −1
ϕ(n) = n ∏ 1 −
= ∏ pj j (pj − 1)
pj
j=1
j=1
Proof. Induction on s. The base case s = 1 is an earlier theorem
in this presentation.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
j=1
s s
1
a −1
ϕ(n) = n ∏ 1 −
= ∏ pj j (pj − 1)
pj
j=1
j=1
Proof. Induction on s. The base case s = 1 is an earlier theorem
in this presentation.
Induction step, s → s + 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
j=1
s s
1
a −1
ϕ(n) = n ∏ 1 −
= ∏ pj j (pj − 1)
pj
j=1
j=1
Proof. Induction on s. The base case s = 1 is an earlier theorem
in this presentation.
Induction step, s → s + 1.
!
s+1
ϕ
a
∏ pj j
j=1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
j=1
s s
1
a −1
ϕ(n) = n ∏ 1 −
= ∏ pj j (pj − 1)
pj
j=1
j=1
Proof. Induction on s. The base case s = 1 is an earlier theorem
in this presentation.
Induction step, s → s + 1.
!
!
!
s+1
ϕ
a
∏ pj j
j=1
Bernd Schröder
Multiplicative Functions
s
=ϕ
a
∏ pj j
a
s+1
ps+1
j=1
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
j=1
s s
1
a −1
ϕ(n) = n ∏ 1 −
= ∏ pj j (pj − 1)
pj
j=1
j=1
Proof. Induction on s. The base case s = 1 is an earlier theorem
in this presentation.
Induction step, s → s + 1.
!
!
!
!
s+1
s
s
a
a
a
as+1
as+1 ϕ ∏ pj j = ϕ
= ϕ ∏ pj j ϕ ps+1
∏ pj j ps+1
j=1
Bernd Schröder
Multiplicative Functions
j=1
j=1
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
j=1
s s
1
a −1
ϕ(n) = n ∏ 1 −
= ∏ pj j (pj − 1)
pj
j=1
j=1
Proof. Induction on s. The base case s = 1 is an earlier theorem
in this presentation.
Induction step, s → s + 1.
!
!
!
!
s+1
s
s
a
a
a
as+1
as+1 ϕ ∏ pj j = ϕ
= ϕ ∏ pj j ϕ ps+1
∏ pj j ps+1
j=1
j=1
"
=
s
a −1
∏ pj j
j=1
#
a
s+1
(pj − 1) ps+1
−1
(ps+1 − 1)
j=1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
j=1
s s
1
a −1
ϕ(n) = n ∏ 1 −
= ∏ pj j (pj − 1)
pj
j=1
j=1
Proof. Induction on s. The base case s = 1 is an earlier theorem
in this presentation.
Induction step, s → s + 1.
!
!
!
!
s+1
s
s
a
a
a
as+1
as+1 ϕ ∏ pj j = ϕ
= ϕ ∏ pj j ϕ ps+1
∏ pj j ps+1
j=1
j=1
"
=
s
∏
j=1
Bernd Schröder
Multiplicative Functions
j=1
#
a −1
pj j (pj − 1)
as+1 −1
ps+1
(ps+1 − 1)
s+1
a −1
= ∏ pj j
(pj − 1)
j=1
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
j=1
s s
1
a −1
ϕ(n) = n ∏ 1 −
= ∏ pj j (pj − 1)
pj
j=1
j=1
Proof. Induction on s. The base case s = 1 is an earlier theorem
in this presentation.
Induction step, s → s + 1.
!
!
!
!
s+1
s
s
a
a
a
as+1
as+1 ϕ ∏ pj j = ϕ
= ϕ ∏ pj j ϕ ps+1
∏ pj j ps+1
j=1
j=1
"
=
s
∏
j=1
Bernd Schröder
Multiplicative Functions
j=1
#
a −1
pj j (pj − 1)
as+1 −1
ps+1
(ps+1 − 1)
s+1
a −1
= ∏ pj j
(pj − 1)
j=1
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let n be an integer greater than 2. Then ϕ(n) is
even.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let n be an integer greater than 2. Then ϕ(n) is
even.
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let n be an integer greater than 2. Then ϕ(n) is
even.
s
a
Proof. Let n = ∏ pj j be the prime factorization of n.
j=1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let n be an integer greater than 2. Then ϕ(n) is
even.
s
a
Proof. Let n = ∏ pj j be the prime factorization of n. Then
j=1
s
a −1
ϕ(n) = ∏ pj j
(pj − 1).
j=1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let n be an integer greater than 2. Then ϕ(n) is
even.
s
a
Proof. Let n = ∏ pj j be the prime factorization of n. Then
j=1
s
a −1
ϕ(n) = ∏ pj j
(pj − 1). If n has an odd prime factor pj , the
j=1
product for ϕ(n) has an even factor pj − 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let n be an integer greater than 2. Then ϕ(n) is
even.
s
a
Proof. Let n = ∏ pj j be the prime factorization of n. Then
j=1
s
a −1
ϕ(n) = ∏ pj j
(pj − 1). If n has an odd prime factor pj , the
j=1
product for ϕ(n) has an even factor pj − 1. If n = 2k , then
2k−1 ≥ 2 is a factor in the product for ϕ(n).
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let n be an integer greater than 2. Then ϕ(n) is
even.
s
a
Proof. Let n = ∏ pj j be the prime factorization of n. Then
j=1
s
a −1
ϕ(n) = ∏ pj j
(pj − 1). If n has an odd prime factor pj , the
j=1
product for ϕ(n) has an even factor pj − 1. If n = 2k , then
2k−1 ≥ 2 is a factor in the product for ϕ(n).
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let n be a positive integer. Then ∑ ϕ(d) = n.
d|n
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let n be a positive integer. Then ∑ ϕ(d) = n.
d|n
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let n be a positive integer. Then ∑ ϕ(d) = n.
d|n
Proof. For each d|n, let Cd := {m ∈ {1, . . . , n} : (n, m) = d}.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let n be a positive integer. Then ∑ ϕ(d) = n.
d|n
Proof. For each d|n,
[let Cd := {m ∈ {1, . . . , n} : (n, m) = d}.
Then {1, . . . , n} = Cd .
d|n
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let n be a positive integer. Then ∑ ϕ(d) = n.
d|n
Proof. For each d|n,
[let Cd := {m ∈ {1, . . . , n} : (n, m) = d}.
Then {1, . . . , n} = Cd . Now note that
d|n
n
n
o
no n Cd = m ∈ {1, . . . , n} : m = xd, x ∈ 1, . . . ,
, ,x = 1
d
d
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let n be a positive integer. Then ∑ ϕ(d) = n.
d|n
Proof. For each d|n,
[let Cd := {m ∈ {1, . . . , n} : (n, m) = d}.
Then {1, . . . , n} = Cd . Now note that
d|n
n
n
o
no n Cd = m ∈ {1, . . . , n} : m = xd, x ∈ 1, . . . ,
, ,x = 1 ,
d
d
n
so |Cd | = ϕ
.
d
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let n be a positive integer. Then ∑ ϕ(d) = n.
d|n
Proof. For each d|n,
[let Cd := {m ∈ {1, . . . , n} : (n, m) = d}.
Then {1, . . . , n} = Cd . Now note that
d|n
n
n
o
no n Cd = m ∈ {1, . . . , n} : m = xd, x ∈ 1, . . . ,
, ,x = 1 ,
d
d
n
so |Cd | = ϕ
. Hence
d
n =
∑ |Cd |
d|n
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let n be a positive integer. Then ∑ ϕ(d) = n.
d|n
Proof. For each d|n,
[let Cd := {m ∈ {1, . . . , n} : (n, m) = d}.
Then {1, . . . , n} = Cd . Now note that
d|n
n
n
o
no n Cd = m ∈ {1, . . . , n} : m = xd, x ∈ 1, . . . ,
, ,x = 1 ,
d
d
n
so |Cd | = ϕ
. Hence
d
n
n = ∑ |Cd | = ∑ ϕ
d
d|n
d|n
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let n be a positive integer. Then ∑ ϕ(d) = n.
d|n
Proof. For each d|n,
[let Cd := {m ∈ {1, . . . , n} : (n, m) = d}.
Then {1, . . . , n} = Cd . Now note that
d|n
n
n
o
no n Cd = m ∈ {1, . . . , n} : m = xd, x ∈ 1, . . . ,
, ,x = 1 ,
d
d
n
so |Cd | = ϕ
. Hence
d
n
n = ∑ |Cd | = ∑ ϕ
d
d|n
d|n
=
∑ ϕ(d).
d|n
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let n be a positive integer. Then ∑ ϕ(d) = n.
d|n
Proof. For each d|n,
[let Cd := {m ∈ {1, . . . , n} : (n, m) = d}.
Then {1, . . . , n} = Cd . Now note that
d|n
n
n
o
no n Cd = m ∈ {1, . . . , n} : m = xd, x ∈ 1, . . . ,
, ,x = 1 ,
d
d
n
so |Cd | = ϕ
. Hence
d
n
n = ∑ |Cd | = ∑ ϕ
d
d|n
d|n
=
∑ ϕ(d).
d|n
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Example.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Example. Find all solutions of the equation ϕ(n) = 7.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Example. Find all solutions of the equation ϕ(n) = 7.
Because 7 is not even and ϕ(2) = 1, there is no solution.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Example. Find all solutions of the equation ϕ(n) = 7.
Because 7 is not even and ϕ(2) = 1, there is no solution.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Example. Find all solutions of the equation ϕ(n) = 7.
Because 7 is not even and ϕ(2) = 1, there is no solution.
For even right sides, the argument gets more complicated.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The sum of divisors function σ assigns to every
positive integer the sum σ (n) of all its divisors (including 1).
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The sum of divisors function σ assigns to every
positive integer the sum σ (n) of all its divisors (including 1).
Definition.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The sum of divisors function σ assigns to every
positive integer the sum σ (n) of all its divisors (including 1).
Definition. The number of divisors function τ assigns to every
positive integer the number τ(n) of all its divisors (including 1).
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The sum of divisors function σ assigns to every
positive integer the sum σ (n) of all its divisors (including 1).
Definition. The number of divisors function τ assigns to every
positive integer the number τ(n) of all its divisors (including 1).
So σ (n) = ∑ d and τ(n) = ∑ 1.
d|n
Bernd Schröder
Multiplicative Functions
d|n
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The sum of divisors function σ assigns to every
positive integer the sum σ (n) of all its divisors (including 1).
Definition. The number of divisors function τ assigns to every
positive integer the number τ(n) of all its divisors (including 1).
So σ (n) = ∑ d and τ(n) = ∑ 1. That is, both σ and τ are
d|n
d|n
summary functions
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If f is multiplicative, then F(n) := ∑ f (d) is
d|n
multiplicative, too.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If f is multiplicative, then F(n) := ∑ f (d) is
d|n
multiplicative, too.
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If f is multiplicative, then F(n) := ∑ f (d) is
d|n
multiplicative, too.
Proof. Let m, n ∈ N be so that (m, n) = 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If f is multiplicative, then F(n) := ∑ f (d) is
d|n
multiplicative, too.
Proof. Let m, n ∈ N be so that (m, n) = 1. Then every x ∈ N so
that x|mn has a unique factorization x = de, where d, e ∈ N are
so that d|m and e|n.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If f is multiplicative, then F(n) := ∑ f (d) is
d|n
multiplicative, too.
Proof. Let m, n ∈ N be so that (m, n) = 1. Then every x ∈ N so
that x|mn has a unique factorization x = de, where d, e ∈ N are
so that d|m and e|n. Hence
F(m)F(n)
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If f is multiplicative, then F(n) := ∑ f (d) is
d|n
multiplicative, too.
Proof. Let m, n ∈ N be so that (m, n) = 1. Then every x ∈ N so
that x|mn has a unique factorization x = de, where d, e ∈ N are
so that d|m and e|n. Hence
F(m)F(n) =
∑ f (d) ∑ f (e)
d|m
Bernd Schröder
Multiplicative Functions
e|n
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If f is multiplicative, then F(n) := ∑ f (d) is
d|n
multiplicative, too.
Proof. Let m, n ∈ N be so that (m, n) = 1. Then every x ∈ N so
that x|mn has a unique factorization x = de, where d, e ∈ N are
so that d|m and e|n. Hence
F(m)F(n) =
∑ f (d) ∑ f (e) = ∑ ∑ f (d)f (e)
d|m
Bernd Schröder
Multiplicative Functions
e|n
d|m e|n
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If f is multiplicative, then F(n) := ∑ f (d) is
d|n
multiplicative, too.
Proof. Let m, n ∈ N be so that (m, n) = 1. Then every x ∈ N so
that x|mn has a unique factorization x = de, where d, e ∈ N are
so that d|m and e|n. Hence
F(m)F(n) =
∑ f (d) ∑ f (e) = ∑ ∑ f (d)f (e) = ∑
d|m
Bernd Schröder
Multiplicative Functions
e|n
d|m e|n
f (de)
d|m,e|n
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If f is multiplicative, then F(n) := ∑ f (d) is
d|n
multiplicative, too.
Proof. Let m, n ∈ N be so that (m, n) = 1. Then every x ∈ N so
that x|mn has a unique factorization x = de, where d, e ∈ N are
so that d|m and e|n. Hence
F(m)F(n) =
∑ f (d) ∑ f (e) = ∑ ∑ f (d)f (e) = ∑
d|m
=
e|n
d|m e|n
f (de)
d|m,e|n
∑ f (x)
x|mn
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If f is multiplicative, then F(n) := ∑ f (d) is
d|n
multiplicative, too.
Proof. Let m, n ∈ N be so that (m, n) = 1. Then every x ∈ N so
that x|mn has a unique factorization x = de, where d, e ∈ N are
so that d|m and e|n. Hence
F(m)F(n) =
∑ f (d) ∑ f (e) = ∑ ∑ f (d)f (e) = ∑
d|m
=
e|n
d|m e|n
f (de)
d|m,e|n
∑ f (x) = F(mn).
x|mn
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If f is multiplicative, then F(n) := ∑ f (d) is
d|n
multiplicative, too.
Proof. Let m, n ∈ N be so that (m, n) = 1. Then every x ∈ N so
that x|mn has a unique factorization x = de, where d, e ∈ N are
so that d|m and e|n. Hence
F(m)F(n) =
∑ f (d) ∑ f (e) = ∑ ∑ f (d)f (e) = ∑
d|m
=
e|n
d|m e|n
f (de)
d|m,e|n
∑ f (x) = F(mn).
x|mn
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If f is multiplicative, then F(n) := ∑ f (d) is
d|n
multiplicative, too.
Proof. Let m, n ∈ N be so that (m, n) = 1. Then every x ∈ N so
that x|mn has a unique factorization x = de, where d, e ∈ N are
so that d|m and e|n. Hence
F(m)F(n) =
∑ f (d) ∑ f (e) = ∑ ∑ f (d)f (e) = ∑
d|m
=
e|n
d|m e|n
f (de)
d|m,e|n
∑ f (x) = F(mn).
x|mn
Theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If f is multiplicative, then F(n) := ∑ f (d) is
d|n
multiplicative, too.
Proof. Let m, n ∈ N be so that (m, n) = 1. Then every x ∈ N so
that x|mn has a unique factorization x = de, where d, e ∈ N are
so that d|m and e|n. Hence
F(m)F(n) =
∑ f (d) ∑ f (e) = ∑ ∑ f (d)f (e) = ∑
d|m
=
e|n
d|m e|n
f (de)
d|m,e|n
∑ f (x) = F(mn).
x|mn
Theorem. σ and τ are multiplicative.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If f is multiplicative, then F(n) := ∑ f (d) is
d|n
multiplicative, too.
Proof. Let m, n ∈ N be so that (m, n) = 1. Then every x ∈ N so
that x|mn has a unique factorization x = de, where d, e ∈ N are
so that d|m and e|n. Hence
F(m)F(n) =
∑ f (d) ∑ f (e) = ∑ ∑ f (d)f (e) = ∑
d|m
=
e|n
d|m e|n
f (de)
d|m,e|n
∑ f (x) = F(mn).
x|mn
Theorem. σ and τ are multiplicative.
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If f is multiplicative, then F(n) := ∑ f (d) is
d|n
multiplicative, too.
Proof. Let m, n ∈ N be so that (m, n) = 1. Then every x ∈ N so
that x|mn has a unique factorization x = de, where d, e ∈ N are
so that d|m and e|n. Hence
F(m)F(n) =
∑ f (d) ∑ f (e) = ∑ ∑ f (d)f (e) = ∑
d|m
=
e|n
d|m e|n
f (de)
d|m,e|n
∑ f (x) = F(mn).
x|mn
Theorem. σ and τ are multiplicative.
Proof. Use f (n) = n and f (n) = 1 in the preceding theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If f is multiplicative, then F(n) := ∑ f (d) is
d|n
multiplicative, too.
Proof. Let m, n ∈ N be so that (m, n) = 1. Then every x ∈ N so
that x|mn has a unique factorization x = de, where d, e ∈ N are
so that d|m and e|n. Hence
F(m)F(n) =
∑ f (d) ∑ f (e) = ∑ ∑ f (d)f (e) = ∑
d|m
=
e|n
d|m e|n
f (de)
d|m,e|n
∑ f (x) = F(mn).
x|mn
Theorem. σ and τ are multiplicative.
Proof. Use f (n) = n and f (n) = 1 in the preceding theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Lemma.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Lemma. If p is prime and a is a positive integer, then
a
pa+1 − 1
σ (pa ) = ∑ pj =
.
p−1
j=0
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Lemma. If p is prime and a is a positive integer, then
a
pa+1 − 1
σ (pa ) = ∑ pj =
.
p−1
j=0
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Lemma. If p is prime and a is a positive integer, then
a
pa+1 − 1
σ (pa ) = ∑ pj =
.
p−1
j=0
Proof. The divisors of pa are 1, p, p2 , . . . , pa .
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Lemma. If p is prime and a is a positive integer, then
a
pa+1 − 1
σ (pa ) = ∑ pj =
.
p−1
j=0
Proof. The divisors of pa are 1, p, p2 , . . . , pa . Hence
σ (pa )
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Lemma. If p is prime and a is a positive integer, then
a
pa+1 − 1
σ (pa ) = ∑ pj =
.
p−1
j=0
Proof. The divisors of pa are 1, p, p2 , . . . , pa . Hence
a
σ (pa ) = ∑ pj
j=0
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Lemma. If p is prime and a is a positive integer, then
a
pa+1 − 1
σ (pa ) = ∑ pj =
.
p−1
j=0
Proof. The divisors of pa are 1, p, p2 , . . . , pa . Hence
a
σ (pa ) = ∑ pj =
j=0
Bernd Schröder
Multiplicative Functions
pa+1 − 1
.
p−1
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Lemma. If p is prime and a is a positive integer, then
a
pa+1 − 1
σ (pa ) = ∑ pj =
.
p−1
j=0
Proof. The divisors of pa are 1, p, p2 , . . . , pa . Hence
a
σ (pa ) = ∑ pj =
j=0
Bernd Schröder
Multiplicative Functions
pa+1 − 1
.
p−1
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
s
σ (n) = ∏
j=1
Bernd Schröder
Multiplicative Functions
j=1
aj +1
pj
−1
pj − 1
s
and τ (n) = ∏(aj + 1).
j=1
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
s
σ (n) = ∏
j=1
j=1
aj +1
pj
−1
pj − 1
s
and τ (n) = ∏(aj + 1).
j=1
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
s
σ (n) = ∏
j=1
j=1
aj +1
pj
−1
pj − 1
s
and τ (n) = ∏(aj + 1).
j=1
Proof. Because (pi , pj ) = 1 for i 6= j
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
s
σ (n) = ∏
j=1
j=1
aj +1
pj
−1
pj − 1
s
and τ (n) = ∏(aj + 1).
j=1
Proof. Because (pi , pj ) = 1 for i 6= j, we have
σ (n)
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
s
σ (n) = ∏
j=1
j=1
aj +1
pj
−1
s
and τ (n) = ∏(aj + 1).
pj − 1
j=1
Proof. Because (pi , pj ) = 1 for i 6= j, we have
!
s
σ (n) = σ
a
∏ pj j
j=1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
s
σ (n) = ∏
j=1
j=1
aj +1
pj
−1
pj − 1
s
and τ (n) = ∏(aj + 1).
j=1
Proof. Because (pi , pj ) = 1 for i 6= j, we have
!
s
s
a
a
σ (n) = σ ∏ pj j = ∏ σ pj j
j=1
Bernd Schröder
Multiplicative Functions
j=1
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
s
σ (n) = ∏
j=1
j=1
aj +1
pj
−1
pj − 1
s
and τ (n) = ∏(aj + 1).
j=1
Proof. Because (pi , pj ) = 1 for i 6= j, we have
!
a +1
s
s p j
s
−1
aj
aj
j
= ∏ σ pj = ∏
.
σ (n) = σ ∏ pj
p
−
1
j
j=1
j=1
j=1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
s
σ (n) = ∏
j=1
j=1
aj +1
pj
−1
pj − 1
s
and τ (n) = ∏(aj + 1).
j=1
Proof. Because (pi , pj ) = 1 for i 6= j, we have
!
a +1
s
s p j
s
−1
aj
aj
j
= ∏ σ pj = ∏
.
σ (n) = σ ∏ pj
p
−
1
j
j=1
j=1
j=1
For τ the same argument, using τ (pa ) = a + 1, will work.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
s
Perfect Numbers and Mersenne Primes
Möbius Inversion
a
Theorem. Let n = ∏ pj j be the prime factorization of n. Then
s
σ (n) = ∏
j=1
j=1
aj +1
pj
−1
pj − 1
s
and τ (n) = ∏(aj + 1).
j=1
Proof. Because (pi , pj ) = 1 for i 6= j, we have
!
a +1
s
s p j
s
−1
aj
aj
j
= ∏ σ pj = ∏
.
σ (n) = σ ∏ pj
p
−
1
j
j=1
j=1
j=1
For τ the same argument, using τ (pa ) = a + 1, will work.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. A number is perfect iff it is the sum of its proper
positive divisors.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. A number is perfect iff it is the sum of its proper
positive divisors. (This is the case iff σ (n) = 2n.)
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. A number is perfect iff it is the sum of its proper
positive divisors. (This is the case iff σ (n) = 2n.)
Theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. A number is perfect iff it is the sum of its proper
positive divisors. (This is the case iff σ (n) = 2n.)
Theorem. The even positive integer n is a perfect number iff
n = 2m−1 (2m − 1), where m is an integer so that m ≥ 2 and
2m − 1 is prime.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. A number is perfect iff it is the sum of its proper
positive divisors. (This is the case iff σ (n) = 2n.)
Theorem. The even positive integer n is a perfect number iff
n = 2m−1 (2m − 1), where m is an integer so that m ≥ 2 and
2m − 1 is prime.
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. A number is perfect iff it is the sum of its proper
positive divisors. (This is the case iff σ (n) = 2n.)
Theorem. The even positive integer n is a perfect number iff
n = 2m−1 (2m − 1), where m is an integer so that m ≥ 2 and
2m − 1 is prime.
Proof. Let n = 2m−1 j, where j is odd and m ≥ 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. A number is perfect iff it is the sum of its proper
positive divisors. (This is the case iff σ (n) = 2n.)
Theorem. The even positive integer n is a perfect number iff
n = 2m−1 (2m − 1), where m is an integer so that m ≥ 2 and
2m − 1 is prime.
Proof. Let n = 2m−1 j, where j is odd and m ≥ 1. Note that
σ (j) ≥ j + 1 with equality holding iff j is prime.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. A number is perfect iff it is the sum of its proper
positive divisors. (This is the case iff σ (n) = 2n.)
Theorem. The even positive integer n is a perfect number iff
n = 2m−1 (2m − 1), where m is an integer so that m ≥ 2 and
2m − 1 is prime.
Proof. Let n = 2m−1 j, where j is odd and m ≥ 1. Note that
σ (j) ≥ j + 1 with equality holding iff j is prime. Also note that
σ (n) = 2n iff
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. A number is perfect iff it is the sum of its proper
positive divisors. (This is the case iff σ (n) = 2n.)
Theorem. The even positive integer n is a perfect number iff
n = 2m−1 (2m − 1), where m is an integer so that m ≥ 2 and
2m − 1 is prime.
Proof. Let n = 2m−1 j, where j is odd and m ≥ 1. Note that
σ (j) ≥ j + 1 with equality holding iff j is prime. Also note that
σ (n) = 2n iff
2m j
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. A number is perfect iff it is the sum of its proper
positive divisors. (This is the case iff σ (n) = 2n.)
Theorem. The even positive integer n is a perfect number iff
n = 2m−1 (2m − 1), where m is an integer so that m ≥ 2 and
2m − 1 is prime.
Proof. Let n = 2m−1 j, where j is odd and m ≥ 1. Note that
σ (j) ≥ j + 1 with equality holding iff j is prime. Also note that
σ (n) = 2n iff
2m j = 2 · 2m−1 j
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. A number is perfect iff it is the sum of its proper
positive divisors. (This is the case iff σ (n) = 2n.)
Theorem. The even positive integer n is a perfect number iff
n = 2m−1 (2m − 1), where m is an integer so that m ≥ 2 and
2m − 1 is prime.
Proof. Let n = 2m−1 j, where j is odd and m ≥ 1. Note that
σ (j) ≥ j + 1 with equality holding iff j is prime. Also note that
σ (n) = 2n iff
2m j = 2 · 2m−1 j = σ 2m−1 j
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. A number is perfect iff it is the sum of its proper
positive divisors. (This is the case iff σ (n) = 2n.)
Theorem. The even positive integer n is a perfect number iff
n = 2m−1 (2m − 1), where m is an integer so that m ≥ 2 and
2m − 1 is prime.
Proof. Let n = 2m−1 j, where j is odd and m ≥ 1. Note that
σ (j) ≥ j + 1 with equality holding iff j is prime. Also note that
σ (n) = 2n iff
2m j = 2 · 2m−1 j = σ 2m−1 j = σ 2m−1 σ (j)
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. A number is perfect iff it is the sum of its proper
positive divisors. (This is the case iff σ (n) = 2n.)
Theorem. The even positive integer n is a perfect number iff
n = 2m−1 (2m − 1), where m is an integer so that m ≥ 2 and
2m − 1 is prime.
Proof. Let n = 2m−1 j, where j is odd and m ≥ 1. Note that
σ (j) ≥ j + 1 with equality holding iff j is prime. Also note that
σ (n) = 2n iff
2m j = 2 · 2m−1 j = σ 2m−1 j = σ 2m−1 σ (j)
= (2m − 1) σ (j).
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. A number is perfect iff it is the sum of its proper
positive divisors. (This is the case iff σ (n) = 2n.)
Theorem. The even positive integer n is a perfect number iff
n = 2m−1 (2m − 1), where m is an integer so that m ≥ 2 and
2m − 1 is prime.
Proof. Let n = 2m−1 j, where j is odd and m ≥ 1. Note that
σ (j) ≥ j + 1 with equality holding iff j is prime. Also note that
σ (n) = 2n iff
2m j = 2 · 2m−1 j = σ 2m−1 j = σ 2m−1 σ (j)
= (2m − 1) σ (j).
Now note, for “⇐,” that, for j = 2m − 1 prime we have
σ (j) = 2m
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. A number is perfect iff it is the sum of its proper
positive divisors. (This is the case iff σ (n) = 2n.)
Theorem. The even positive integer n is a perfect number iff
n = 2m−1 (2m − 1), where m is an integer so that m ≥ 2 and
2m − 1 is prime.
Proof. Let n = 2m−1 j, where j is odd and m ≥ 1. Note that
σ (j) ≥ j + 1 with equality holding iff j is prime. Also note that
σ (n) = 2n iff
2m j = 2 · 2m−1 j = σ 2m−1 j = σ 2m−1 σ (j)
= (2m − 1) σ (j).
Now note, for “⇐,” that, for j = 2m − 1 prime we have
σ (j) = 2m and the equation holds
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. A number is perfect iff it is the sum of its proper
positive divisors. (This is the case iff σ (n) = 2n.)
Theorem. The even positive integer n is a perfect number iff
n = 2m−1 (2m − 1), where m is an integer so that m ≥ 2 and
2m − 1 is prime.
Proof. Let n = 2m−1 j, where j is odd and m ≥ 1. Note that
σ (j) ≥ j + 1 with equality holding iff j is prime. Also note that
σ (n) = 2n iff
2m j = 2 · 2m−1 j = σ 2m−1 j = σ 2m−1 σ (j)
= (2m − 1) σ (j).
Now note, for “⇐,” that, for j = 2m − 1 prime we have
σ (j) = 2m and the equation holds, which means that even
numbers of the given form are perfect.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.).
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j).
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j). Hence
j = (2m − 1) d and σ (j) = 2m e.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j). Hence
j = (2m − 1) d and σ (j) = 2m e. Substituting into the equation,
we obtain 2m (2m − 1) d
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j). Hence
j = (2m − 1) d and σ (j) = 2m e. Substituting into the equation,
we obtain 2m (2m − 1) d = 2m j
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j). Hence
j = (2m − 1) d and σ (j) = 2m e. Substituting into the equation,
we obtain 2m (2m − 1) d = 2m j = (2m − 1) σ (j)
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j). Hence
j = (2m − 1) d and σ (j) = 2m e. Substituting into the equation,
we obtain 2m (2m − 1) d = 2m j = (2m − 1) σ (j) = (2m − 1) 2m e
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j). Hence
j = (2m − 1) d and σ (j) = 2m e. Substituting into the equation,
we obtain 2m (2m − 1) d = 2m j = (2m − 1) σ (j) = (2m − 1) 2m e,
which means d = e.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j). Hence
j = (2m − 1) d and σ (j) = 2m e. Substituting into the equation,
we obtain 2m (2m − 1) d = 2m j = (2m − 1) σ (j) = (2m − 1) 2m e,
which means d = e. So j = (2m − 1) d and, if d > 1, we would
have 2m d = σ (j)
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j). Hence
j = (2m − 1) d and σ (j) = 2m e. Substituting into the equation,
we obtain 2m (2m − 1) d = 2m j = (2m − 1) σ (j) = (2m − 1) 2m e,
which means d = e. So j = (2m − 1) d and, if d > 1, we would
have 2m d = σ (j) ≥ (2m − 1) d + (2m − 1) + d
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j). Hence
j = (2m − 1) d and σ (j) = 2m e. Substituting into the equation,
we obtain 2m (2m − 1) d = 2m j = (2m − 1) σ (j) = (2m − 1) 2m e,
which means d = e. So j = (2m − 1) d and, if d > 1, we would
have 2m d = σ (j) ≥ (2m − 1) d + (2m − 1) + d = 2m d + (2m − 1)
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j). Hence
j = (2m − 1) d and σ (j) = 2m e. Substituting into the equation,
we obtain 2m (2m − 1) d = 2m j = (2m − 1) σ (j) = (2m − 1) 2m e,
which means d = e. So j = (2m − 1) d and, if d > 1, we would
have 2m d = σ (j) ≥ (2m − 1) d + (2m − 1) + d = 2m d + (2m − 1),
which is not possible.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j). Hence
j = (2m − 1) d and σ (j) = 2m e. Substituting into the equation,
we obtain 2m (2m − 1) d = 2m j = (2m − 1) σ (j) = (2m − 1) 2m e,
which means d = e. So j = (2m − 1) d and, if d > 1, we would
have 2m d = σ (j) ≥ (2m − 1) d + (2m − 1) + d = 2m d + (2m − 1),
which is not possible. Thus d = 1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j). Hence
j = (2m − 1) d and σ (j) = 2m e. Substituting into the equation,
we obtain 2m (2m − 1) d = 2m j = (2m − 1) σ (j) = (2m − 1) 2m e,
which means d = e. So j = (2m − 1) d and, if d > 1, we would
have 2m d = σ (j) ≥ (2m − 1) d + (2m − 1) + d = 2m d + (2m − 1),
which is not possible. Thus d = 1 and j = (2m − 1)
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j). Hence
j = (2m − 1) d and σ (j) = 2m e. Substituting into the equation,
we obtain 2m (2m − 1) d = 2m j = (2m − 1) σ (j) = (2m − 1) 2m e,
which means d = e. So j = (2m − 1) d and, if d > 1, we would
have 2m d = σ (j) ≥ (2m − 1) d + (2m − 1) + d = 2m d + (2m − 1),
which is not possible. Thus d = 1 and j = (2m − 1) and
σ (j) = 2m
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j). Hence
j = (2m − 1) d and σ (j) = 2m e. Substituting into the equation,
we obtain 2m (2m − 1) d = 2m j = (2m − 1) σ (j) = (2m − 1) 2m e,
which means d = e. So j = (2m − 1) d and, if d > 1, we would
have 2m d = σ (j) ≥ (2m − 1) d + (2m − 1) + d = 2m d + (2m − 1),
which is not possible. Thus d = 1 and j = (2m − 1) and
σ (j) = 2m = j + 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j). Hence
j = (2m − 1) d and σ (j) = 2m e. Substituting into the equation,
we obtain 2m (2m − 1) d = 2m j = (2m − 1) σ (j) = (2m − 1) 2m e,
which means d = e. So j = (2m − 1) d and, if d > 1, we would
have 2m d = σ (j) ≥ (2m − 1) d + (2m − 1) + d = 2m d + (2m − 1),
which is not possible. Thus d = 1 and j = (2m − 1) and
σ (j) = 2m = j + 1. Now, if j was not prime, we would have
σ (j) > j + 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j). Hence
j = (2m − 1) d and σ (j) = 2m e. Substituting into the equation,
we obtain 2m (2m − 1) d = 2m j = (2m − 1) σ (j) = (2m − 1) 2m e,
which means d = e. So j = (2m − 1) d and, if d > 1, we would
have 2m d = σ (j) ≥ (2m − 1) d + (2m − 1) + d = 2m d + (2m − 1),
which is not possible. Thus d = 1 and j = (2m − 1) and
σ (j) = 2m = j + 1. Now, if j was not prime, we would have
σ (j) > j + 1. Thus j = (2m − 1) is prime and n is as indicated.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Proof (cont.). For “⇒,” first note that the equation
2m j = (2m − 1) σ (j) implies that 2m − 1|j and 2m |σ (j). Hence
j = (2m − 1) d and σ (j) = 2m e. Substituting into the equation,
we obtain 2m (2m − 1) d = 2m j = (2m − 1) σ (j) = (2m − 1) 2m e,
which means d = e. So j = (2m − 1) d and, if d > 1, we would
have 2m d = σ (j) ≥ (2m − 1) d + (2m − 1) + d = 2m d + (2m − 1),
which is not possible. Thus d = 1 and j = (2m − 1) and
σ (j) = 2m = j + 1. Now, if j was not prime, we would have
σ (j) > j + 1. Thus j = (2m − 1) is prime and n is as indicated.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If m is a positive integer and 2m − 1 is prime, then m
must be prime.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If m is a positive integer and 2m − 1 is prime, then m
must be prime.
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If m is a positive integer and 2m − 1 is prime, then m
must be prime.
Proof. For m = pq with p, q ∈ N \ {1} we have that
2m − 1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If m is a positive integer and 2m − 1 is prime, then m
must be prime.
Proof. For m = pq with p, q ∈ N \ {1} we have that
2m − 1 = (2p )q − 1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If m is a positive integer and 2m − 1 is prime, then m
must be prime.
Proof. For m = pq with p, q ∈ N \ {1} we have that
q−1
2 − 1 = (2 ) − 1 = (2 − 1) ∑ (2p )j
m
p q
p
j=0
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If m is a positive integer and 2m − 1 is prime, then m
must be prime.
Proof. For m = pq with p, q ∈ N \ {1} we have that
q−1
2 − 1 = (2 ) − 1 = (2 − 1) ∑ (2p )j ,
m
p q
p
j=0
which is composite.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If m is a positive integer and 2m − 1 is prime, then m
must be prime.
Proof. For m = pq with p, q ∈ N \ {1} we have that
q−1
2 − 1 = (2 ) − 1 = (2 − 1) ∑ (2p )j ,
m
p q
p
j=0
which is composite.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If m is a positive integer and 2m − 1 is prime, then m
must be prime.
Proof. For m = pq with p, q ∈ N \ {1} we have that
q−1
2 − 1 = (2 ) − 1 = (2 − 1) ∑ (2p )j ,
m
p q
p
j=0
which is composite.
Definition.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If m is a positive integer and 2m − 1 is prime, then m
must be prime.
Proof. For m = pq with p, q ∈ N \ {1} we have that
q−1
2 − 1 = (2 ) − 1 = (2 − 1) ∑ (2p )j ,
m
p q
p
j=0
which is composite.
Definition. The number Mm := 2m − 1 is the mth Mersenne
number. If Mm is prime, it is called a Mersenne prime.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If m is a positive integer and 2m − 1 is prime, then m
must be prime.
Proof. For m = pq with p, q ∈ N \ {1} we have that
q−1
2 − 1 = (2 ) − 1 = (2 − 1) ∑ (2p )j ,
m
p q
p
j=0
which is composite.
Definition. The number Mm := 2m − 1 is the mth Mersenne
number. If Mm is prime, it is called a Mersenne prime.
The theorem shows that not all Mersenne numbers Mm are
prime.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If m is a positive integer and 2m − 1 is prime, then m
must be prime.
Proof. For m = pq with p, q ∈ N \ {1} we have that
q−1
2 − 1 = (2 ) − 1 = (2 − 1) ∑ (2p )j ,
m
p q
p
j=0
which is composite.
Definition. The number Mm := 2m − 1 is the mth Mersenne
number. If Mm is prime, it is called a Mersenne prime.
The theorem shows that not all Mersenne numbers Mm are
prime. But even when m is prime, Mm need not be prime.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If m is a positive integer and 2m − 1 is prime, then m
must be prime.
Proof. For m = pq with p, q ∈ N \ {1} we have that
q−1
2 − 1 = (2 ) − 1 = (2 − 1) ∑ (2p )j ,
m
p q
p
j=0
which is composite.
Definition. The number Mm := 2m − 1 is the mth Mersenne
number. If Mm is prime, it is called a Mersenne prime.
The theorem shows that not all Mersenne numbers Mm are
prime. But even when m is prime, Mm need not be prime.
M11 = 211 − 1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If m is a positive integer and 2m − 1 is prime, then m
must be prime.
Proof. For m = pq with p, q ∈ N \ {1} we have that
q−1
2 − 1 = (2 ) − 1 = (2 − 1) ∑ (2p )j ,
m
p q
p
j=0
which is composite.
Definition. The number Mm := 2m − 1 is the mth Mersenne
number. If Mm is prime, it is called a Mersenne prime.
The theorem shows that not all Mersenne numbers Mm are
prime. But even when m is prime, Mm need not be prime.
M11 = 211 − 1 = 2047
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. If m is a positive integer and 2m − 1 is prime, then m
must be prime.
Proof. For m = pq with p, q ∈ N \ {1} we have that
q−1
2 − 1 = (2 ) − 1 = (2 − 1) ∑ (2p )j ,
m
p q
p
j=0
which is composite.
Definition. The number Mm := 2m − 1 is the mth Mersenne
number. If Mm is prime, it is called a Mersenne prime.
The theorem shows that not all Mersenne numbers Mm are
prime. But even when m is prime, Mm need not be prime.
M11 = 211 − 1 = 2047 = 23 · 89.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let p be an odd prime number and let d be a divisor
of the Mersenne number Mp = 2p − 1. Then, for some positive
integer k, we have d = 2kp + 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let p be an odd prime number and let d be a divisor
of the Mersenne number Mp = 2p − 1. Then, for some positive
integer k, we have d = 2kp + 1.
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let p be an odd prime number and let d be a divisor
of the Mersenne number Mp = 2p − 1. Then, for some positive
integer k, we have d = 2kp + 1.
Proof. Let q be a prime factor of Mp = 2p − 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let p be an odd prime number and let d be a divisor
of the Mersenne number Mp = 2p − 1. Then, for some positive
integer k, we have d = 2kp + 1.
Proof. Let q be a prime factor of Mp = 2p − 1. By Fermat’s
Little Theorem, q|2q−1 − 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let p be an odd prime number and let d be a divisor
of the Mersenne number Mp = 2p − 1. Then, for some positive
integer k, we have d = 2kp + 1.
Proof. Let q be a prime factor of Mp = 2p − 1. By Fermat’s
Little Theorem, q|2q−1 − 1. Hence 2p − 1, 2q−1 − 1 = q
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let p be an odd prime number and let d be a divisor
of the Mersenne number Mp = 2p − 1. Then, for some positive
integer k, we have d = 2kp + 1.
Proof. Let q be a prime factor of Mp = 2p − 1. By Fermat’s
Little Theorem, q|2q−1 − 1. Hence 2p − 1, 2q−1 − 1 = q,
which, by earlier result, implies that (p, q − 1) 6= 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let p be an odd prime number and let d be a divisor
of the Mersenne number Mp = 2p − 1. Then, for some positive
integer k, we have d = 2kp + 1.
Proof. Let q be a prime factor of Mp = 2p − 1. By Fermat’s
Little Theorem, q|2q−1 − 1. Hence 2p − 1, 2q−1 − 1 = q,
which, by earlier result, implies that (p, q − 1) 6= 1. Therefore,
p|q − 1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let p be an odd prime number and let d be a divisor
of the Mersenne number Mp = 2p − 1. Then, for some positive
integer k, we have d = 2kp + 1.
Proof. Let q be a prime factor of Mp = 2p − 1. By Fermat’s
Little Theorem, q|2q−1 − 1. Hence 2p − 1, 2q−1 − 1 = q,
which, by earlier result, implies that (p, q − 1) 6= 1. Therefore,
p|q − 1, that is, q − 1 = mp.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let p be an odd prime number and let d be a divisor
of the Mersenne number Mp = 2p − 1. Then, for some positive
integer k, we have d = 2kp + 1.
Proof. Let q be a prime factor of Mp = 2p − 1. By Fermat’s
Little Theorem, q|2q−1 − 1. Hence 2p − 1, 2q−1 − 1 = q,
which, by earlier result, implies that (p, q − 1) 6= 1. Therefore,
p|q − 1, that is, q − 1 = mp. Because q is odd, m = 2k for some
k.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let p be an odd prime number and let d be a divisor
of the Mersenne number Mp = 2p − 1. Then, for some positive
integer k, we have d = 2kp + 1.
Proof. Let q be a prime factor of Mp = 2p − 1. By Fermat’s
Little Theorem, q|2q−1 − 1. Hence 2p − 1, 2q−1 − 1 = q,
which, by earlier result, implies that (p, q − 1) 6= 1. Therefore,
p|q − 1, that is, q − 1 = mp. Because q is odd, m = 2k for some
k. Consequently, q = 2kp + 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let p be an odd prime number and let d be a divisor
of the Mersenne number Mp = 2p − 1. Then, for some positive
integer k, we have d = 2kp + 1.
Proof. Let q be a prime factor of Mp = 2p − 1. By Fermat’s
Little Theorem, q|2q−1 − 1. Hence 2p − 1, 2q−1 − 1 = q,
which, by earlier result, implies that (p, q − 1) 6= 1. Therefore,
p|q − 1, that is, q − 1 = mp. Because q is odd, m = 2k for some
k. Consequently, q = 2kp + 1. Because any divisor of Mp is a
product of prime divisors
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let p be an odd prime number and let d be a divisor
of the Mersenne number Mp = 2p − 1. Then, for some positive
integer k, we have d = 2kp + 1.
Proof. Let q be a prime factor of Mp = 2p − 1. By Fermat’s
Little Theorem, q|2q−1 − 1. Hence 2p − 1, 2q−1 − 1 = q,
which, by earlier result, implies that (p, q − 1) 6= 1. Therefore,
p|q − 1, that is, q − 1 = mp. Because q is odd, m = 2k for some
k. Consequently, q = 2kp + 1. Because any divisor of Mp is a
product of prime divisors, a simple induction on the number of
prime factors (with multiplicity) shows that any divisor of Mp is
of the form d = 2kp + 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let p be an odd prime number and let d be a divisor
of the Mersenne number Mp = 2p − 1. Then, for some positive
integer k, we have d = 2kp + 1.
Proof. Let q be a prime factor of Mp = 2p − 1. By Fermat’s
Little Theorem, q|2q−1 − 1. Hence 2p − 1, 2q−1 − 1 = q,
which, by earlier result, implies that (p, q − 1) 6= 1. Therefore,
p|q − 1, that is, q − 1 = mp. Because q is odd, m = 2k for some
k. Consequently, q = 2kp + 1. Because any divisor of Mp is a
product of prime divisors, a simple induction on the number of
prime factors (with multiplicity) shows that any divisor of Mp is
of the form d = 2kp + 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Searching for Mersenne Primes
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Searching for Mersenne
Primes
1. There is an O p3 test whether (for prime p) Mp is prime.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Searching for Mersenne
Primes
1. There is an O p3 test whether (for prime p) Mp is prime.
2. The search for Mersenne primes has a long history and it is
still ongoing. (Now using distributed computing on the
web, see http://mersenne.org/)
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Searching for Mersenne
Primes
1. There is an O p3 test whether (for prime p) Mp is prime.
2. The search for Mersenne primes has a long history and it is
still ongoing. (Now using distributed computing on the
web, see http://mersenne.org/)
3. It has been conjectured, but not proved, that there are
infinitely many Mersenne prime numbers.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Searching for Mersenne
Primes
1. There is an O p3 test whether (for prime p) Mp is prime.
2. The search for Mersenne primes has a long history and it is
still ongoing. (Now using distributed computing on the
web, see http://mersenne.org/)
3. It has been conjectured, but not proved, that there are
infinitely many Mersenne prime numbers.
One motivation for looking for Mersenne primes is the search
for even perfect numbers.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Searching for Mersenne
Primes
1. There is an O p3 test whether (for prime p) Mp is prime.
2. The search for Mersenne primes has a long history and it is
still ongoing. (Now using distributed computing on the
web, see http://mersenne.org/)
3. It has been conjectured, but not proved, that there are
infinitely many Mersenne prime numbers.
One motivation for looking for Mersenne primes is the search
for even perfect numbers. Odd perfect numbers would need to
satisfy so many conditions, that they would be very large and
difficult, if they exist at all.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Searching for Mersenne
Primes
1. There is an O p3 test whether (for prime p) Mp is prime.
2. The search for Mersenne primes has a long history and it is
still ongoing. (Now using distributed computing on the
web, see http://mersenne.org/)
3. It has been conjectured, but not proved, that there are
infinitely many Mersenne prime numbers.
One motivation for looking for Mersenne primes is the search
for even perfect numbers. Odd perfect numbers would need to
satisfy so many conditions, that they would be very large and
difficult, if they exist at all. But a proof that decides their
existence or nonexistence has not been found yet.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1,
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r and to all other n the value µ(n) = 0.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r and to all other n the value µ(n) = 0.
Theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r and to all other n the value µ(n) = 0.
Theorem. The Möbius function is multiplicative.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r and to all other n the value µ(n) = 0.
Theorem. The Möbius function is multiplicative.
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r and to all other n the value µ(n) = 0.
Theorem. The Möbius function is multiplicative.
Proof. Let (m, n) = 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r and to all other n the value µ(n) = 0.
Theorem. The Möbius function is multiplicative.
Proof. Let (m, n) = 1. Multiplicativity is trivial if one of m or n
is 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r and to all other n the value µ(n) = 0.
Theorem. The Möbius function is multiplicative.
Proof. Let (m, n) = 1. Multiplicativity is trivial if one of m or n
is 1. If µ(m) = 0 or µ(n) = 0, then there is a repeated prime
factor in m or n
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r and to all other n the value µ(n) = 0.
Theorem. The Möbius function is multiplicative.
Proof. Let (m, n) = 1. Multiplicativity is trivial if one of m or n
is 1. If µ(m) = 0 or µ(n) = 0, then there is a repeated prime
factor in m or n and hence also in the product mn.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r and to all other n the value µ(n) = 0.
Theorem. The Möbius function is multiplicative.
Proof. Let (m, n) = 1. Multiplicativity is trivial if one of m or n
is 1. If µ(m) = 0 or µ(n) = 0, then there is a repeated prime
factor in m or n and hence also in the product mn. Thus, in this
case µ(mn) = 0
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r and to all other n the value µ(n) = 0.
Theorem. The Möbius function is multiplicative.
Proof. Let (m, n) = 1. Multiplicativity is trivial if one of m or n
is 1. If µ(m) = 0 or µ(n) = 0, then there is a repeated prime
factor in m or n and hence also in the product mn. Thus, in this
case µ(mn) = 0 = µ(m)µ(n).
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r and to all other n the value µ(n) = 0.
Theorem. The Möbius function is multiplicative.
Proof. Let (m, n) = 1. Multiplicativity is trivial if one of m or n
is 1. If µ(m) = 0 or µ(n) = 0, then there is a repeated prime
factor in m or n and hence also in the product mn. Thus, in this
case µ(mn) = 0 = µ(m)µ(n). If µ(m) 6= 0 and µ(n) 6= 0, then
neither m nor n has a repeated prime factor
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r and to all other n the value µ(n) = 0.
Theorem. The Möbius function is multiplicative.
Proof. Let (m, n) = 1. Multiplicativity is trivial if one of m or n
is 1. If µ(m) = 0 or µ(n) = 0, then there is a repeated prime
factor in m or n and hence also in the product mn. Thus, in this
case µ(mn) = 0 = µ(m)µ(n). If µ(m) 6= 0 and µ(n) 6= 0, then
neither m nor n has a repeated prime factor and hence neither
does mn.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r and to all other n the value µ(n) = 0.
Theorem. The Möbius function is multiplicative.
Proof. Let (m, n) = 1. Multiplicativity is trivial if one of m or n
is 1. If µ(m) = 0 or µ(n) = 0, then there is a repeated prime
factor in m or n and hence also in the product mn. Thus, in this
case µ(mn) = 0 = µ(m)µ(n). If µ(m) 6= 0 and µ(n) 6= 0, then
neither m nor n has a repeated prime factor and hence neither
does mn. Let rm be the number of prime factors of m and rn be
the number of prime factors of n.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r and to all other n the value µ(n) = 0.
Theorem. The Möbius function is multiplicative.
Proof. Let (m, n) = 1. Multiplicativity is trivial if one of m or n
is 1. If µ(m) = 0 or µ(n) = 0, then there is a repeated prime
factor in m or n and hence also in the product mn. Thus, in this
case µ(mn) = 0 = µ(m)µ(n). If µ(m) 6= 0 and µ(n) 6= 0, then
neither m nor n has a repeated prime factor and hence neither
does mn. Let rm be the number of prime factors of m and rn be
the number of prime factors of n. Then
µ(mn) = (−1)rm +rn
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r and to all other n the value µ(n) = 0.
Theorem. The Möbius function is multiplicative.
Proof. Let (m, n) = 1. Multiplicativity is trivial if one of m or n
is 1. If µ(m) = 0 or µ(n) = 0, then there is a repeated prime
factor in m or n and hence also in the product mn. Thus, in this
case µ(mn) = 0 = µ(m)µ(n). If µ(m) 6= 0 and µ(n) 6= 0, then
neither m nor n has a repeated prime factor and hence neither
does mn. Let rm be the number of prime factors of m and rn be
the number of prime factors of n. Then
µ(mn) = (−1)rm +rn = (−1)rm (−1)rn
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r and to all other n the value µ(n) = 0.
Theorem. The Möbius function is multiplicative.
Proof. Let (m, n) = 1. Multiplicativity is trivial if one of m or n
is 1. If µ(m) = 0 or µ(n) = 0, then there is a repeated prime
factor in m or n and hence also in the product mn. Thus, in this
case µ(mn) = 0 = µ(m)µ(n). If µ(m) 6= 0 and µ(n) 6= 0, then
neither m nor n has a repeated prime factor and hence neither
does mn. Let rm be the number of prime factors of m and rn be
the number of prime factors of n. Then
µ(mn) = (−1)rm +rn = (−1)rm (−1)rn = µ(m)µ(n).
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Definition. The Möbius function µ assigns to n = 1 the value
µ(1) := 1, to a product n = p1 · · · pr of r distinct primes the
value µ(n) = (−1)r and to all other n the value µ(n) = 0.
Theorem. The Möbius function is multiplicative.
Proof. Let (m, n) = 1. Multiplicativity is trivial if one of m or n
is 1. If µ(m) = 0 or µ(n) = 0, then there is a repeated prime
factor in m or n and hence also in the product mn. Thus, in this
case µ(mn) = 0 = µ(m)µ(n). If µ(m) 6= 0 and µ(n) 6= 0, then
neither m nor n has a repeated prime factor and hence neither
does mn. Let rm be the number of prime factors of m and rn be
the number of prime factors of n. Then
µ(mn) = (−1)rm +rn = (−1)rm (−1)rn = µ(m)µ(n).
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. The summary function F(n) := ∑ µ(d) satisfies
d|n
∑ µ(d) =
d|n
Bernd Schröder
Multiplicative Functions
1; if n = 1,
0; if n > 1.
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. The summary function F(n) := ∑ µ(d) satisfies
d|n
∑ µ(d) =
d|n
1; if n = 1,
0; if n > 1.
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. The summary function F(n) := ∑ µ(d) satisfies
d|n
∑ µ(d) =
d|n
1; if n = 1,
0; if n > 1.
Proof. By earlier theorem, F is multiplicative, so we can
concentrate on the prime factorization of n.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. The summary function F(n) := ∑ µ(d) satisfies
d|n
∑ µ(d) =
d|n
1; if n = 1,
0; if n > 1.
Proof. By earlier theorem, F is multiplicative, so we can
concentrate on the prime factorization of n. First note that, for
k ∈ N and p prime, we have
F pk
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. The summary function F(n) := ∑ µ(d) satisfies
d|n
∑ µ(d) =
d|n
1; if n = 1,
0; if n > 1.
Proof. By earlier theorem, F is multiplicative, so we can
concentrate on the prime factorization of n. First note that, for
k ∈ N and p prime, we have
F pk = ∑ µ(d)
d|pk
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. The summary function F(n) := ∑ µ(d) satisfies
d|n
∑ µ(d) =
d|n
1; if n = 1,
0; if n > 1.
Proof. By earlier theorem, F is multiplicative, so we can
concentrate on the prime factorization of n. First note that, for
k ∈ N and p prime, we have
k
F pk = ∑ µ(d) = ∑ µ pj
d|pk
Bernd Schröder
Multiplicative Functions
j=0
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. The summary function F(n) := ∑ µ(d) satisfies
d|n
∑ µ(d) =
d|n
1; if n = 1,
0; if n > 1.
Proof. By earlier theorem, F is multiplicative, so we can
concentrate on the prime factorization of n. First note that, for
k ∈ N and p prime, we have
k
k
F pk = ∑ µ(d) = ∑ µ pj = 1 + (−1)1 + ∑ µ pj
d|pk
Bernd Schröder
Multiplicative Functions
j=0
j=2
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. The summary function F(n) := ∑ µ(d) satisfies
d|n
∑ µ(d) =
d|n
1; if n = 1,
0; if n > 1.
Proof. By earlier theorem, F is multiplicative, so we can
concentrate on the prime factorization of n. First note that, for
k ∈ N and p prime, we have
k
k
F pk = ∑ µ(d) = ∑ µ pj = 1 + (−1)1 + ∑ µ pj = 0.
d|pk
Bernd Schröder
Multiplicative Functions
j=0
j=2
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. The summary function F(n) := ∑ µ(d) satisfies
d|n
∑ µ(d) =
d|n
1; if n = 1,
0; if n > 1.
Proof. By earlier theorem, F is multiplicative, so we can
concentrate on the prime factorization of n. First note that, for
k ∈ N and p prime, we have
k
k
F pk = ∑ µ(d) = ∑ µ pj = 1 + (−1)1 + ∑ µ pj = 0.
d|pk
j=0
s
j=2
a
Now, with n = ∏ pj j being the prime factorization of n > 1,
j=1
F(n)
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. The summary function F(n) := ∑ µ(d) satisfies
d|n
1; if n = 1,
0; if n > 1.
∑ µ(d) =
d|n
Proof. By earlier theorem, F is multiplicative, so we can
concentrate on the prime factorization of n. First note that, for
k ∈ N and p prime, we have
k
k
F pk = ∑ µ(d) = ∑ µ pj = 1 + (−1)1 + ∑ µ pj = 0.
d|pk
j=0
s
j=2
a
Now, with n = ∏ pj j being the prime factorization of n > 1,
j=1
!
s
F(n) = F
a
∏ pj j
j=1
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. The summary function F(n) := ∑ µ(d) satisfies
d|n
∑ µ(d) =
d|n
1; if n = 1,
0; if n > 1.
Proof. By earlier theorem, F is multiplicative, so we can
concentrate on the prime factorization of n. First note that, for
k ∈ N and p prime, we have
k
k
F pk = ∑ µ(d) = ∑ µ pj = 1 + (−1)1 + ∑ µ pj = 0.
d|pk
j=0
s
j=2
a
Now, with n = ∏ pj j being the prime factorization of n > 1,
j=1
!
s
s
aj
a
= ∏ F pj j
F(n) = F ∏ pj
j=1
Bernd Schröder
Multiplicative Functions
j=1
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. The summary function F(n) := ∑ µ(d) satisfies
d|n
∑ µ(d) =
d|n
1; if n = 1,
0; if n > 1.
Proof. By earlier theorem, F is multiplicative, so we can
concentrate on the prime factorization of n. First note that, for
k ∈ N and p prime, we have
k
k
F pk = ∑ µ(d) = ∑ µ pj = 1 + (−1)1 + ∑ µ pj = 0.
d|pk
j=0
s
j=2
a
Now, with n = ∏ pj j being the prime factorization of n > 1,
j=1
!
s
s
aj
a
= ∏ F pj j = 0.
F(n) = F ∏ pj
j=1
Bernd Schröder
Multiplicative Functions
j=1
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. The summary function F(n) := ∑ µ(d) satisfies
d|n
∑ µ(d) =
d|n
1; if n = 1,
0; if n > 1.
Proof. By earlier theorem, F is multiplicative, so we can
concentrate on the prime factorization of n. First note that, for
k ∈ N and p prime, we have
k
k
F pk = ∑ µ(d) = ∑ µ pj = 1 + (−1)1 + ∑ µ pj = 0.
d|pk
j=0
s
j=2
a
Now, with n = ∏ pj j being the prime factorization of n > 1,
j=1
!
s
s
aj
a
= ∏ F pj j = 0.
F(n) = F ∏ pj
j=1
Bernd Schröder
Multiplicative Functions
j=1
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. (Möbius Inversion Formula.) Let f be an arithmetic
n
function and let F(n) := ∑ f (d). Then f (n) = ∑ µ(d)F
d
d|n
d|n
for all n.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. (Möbius Inversion Formula.) Let f be an arithmetic
n
function and let F(n) := ∑ f (d). Then f (n) = ∑ µ(d)F
d
d|n
d|n
for all n.
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. (Möbius Inversion Formula.) Let f be an arithmetic
n
function and let F(n) := ∑ f (d). Then f (n) = ∑ µ(d)F
d
d|n
d|n
for all n.
Proof.
∑ µ(d)F
d|n
Bernd Schröder
Multiplicative Functions
n
d
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. (Möbius Inversion Formula.) Let f be an arithmetic
n
function and let F(n) := ∑ f (d). Then f (n) = ∑ µ(d)F
d
d|n
d|n
for all n.
Proof.
∑ µ(d)F
d|n
Bernd Schröder
Multiplicative Functions
n
d
=
∑ µ(d) ∑n f (e)
d|n
e| d
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. (Möbius Inversion Formula.) Let f be an arithmetic
n
function and let F(n) := ∑ f (d). Then f (n) = ∑ µ(d)F
d
d|n
d|n
for all n.
Proof.
∑ µ(d)F
d|n
Bernd Schröder
Multiplicative Functions
n
d
=
∑ µ(d) ∑n f (e) = ∑ ∑n µ(d)f (e)
d|n
e| d
d|n e|
d
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. (Möbius Inversion Formula.) Let f be an arithmetic
n
function and let F(n) := ∑ f (d). Then f (n) = ∑ µ(d)F
d
d|n
d|n
for all n.
Proof.
∑ µ(d)F
d|n
n
d
=
∑ µ(d) ∑n f (e) = ∑ ∑n µ(d)f (e)
=
Multiplicative Functions
d|n e|
d
∑ ∑n µ(d)f (e)
e|n d|
Bernd Schröder
e| d
d|n
e
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. (Möbius Inversion Formula.) Let f be an arithmetic
n
function and let F(n) := ∑ f (d). Then f (n) = ∑ µ(d)F
d
d|n
d|n
for all n.
Proof.
∑ µ(d)F
d|n
n
d
=
∑ µ(d) ∑n f (e) = ∑ ∑n µ(d)f (e)
=
Multiplicative Functions
d|n e|
d
∑ ∑n µ(d)f (e) = ∑ f (e) ∑n µ(d)
e|n d|
Bernd Schröder
e| d
d|n
e
e|n
d| e
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. (Möbius Inversion Formula.) Let f be an arithmetic
n
function and let F(n) := ∑ f (d). Then f (n) = ∑ µ(d)F
d
d|n
d|n
for all n.
Proof.
∑ µ(d)F
d|n
n
d
=
∑ µ(d) ∑n f (e) = ∑ ∑n µ(d)f (e)
e| d
d|n
=
d
∑ ∑n µ(d)f (e) = ∑ f (e) ∑n µ(d)
e|n d|
=
d|n e|
e|n
e
d| e
∑ f (e)1e=n
e|n
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. (Möbius Inversion Formula.) Let f be an arithmetic
n
function and let F(n) := ∑ f (d). Then f (n) = ∑ µ(d)F
d
d|n
d|n
for all n.
Proof.
∑ µ(d)F
d|n
n
d
=
∑ µ(d) ∑n f (e) = ∑ ∑n µ(d)f (e)
e| d
d|n
=
d
∑ ∑n µ(d)f (e) = ∑ f (e) ∑n µ(d)
e|n d|
=
d|n e|
e
e|n
d| e
∑ f (e)1e=n = f (n).
e|n
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. (Möbius Inversion Formula.) Let f be an arithmetic
n
function and let F(n) := ∑ f (d). Then f (n) = ∑ µ(d)F
d
d|n
d|n
for all n.
Proof.
∑ µ(d)F
d|n
n
d
=
∑ µ(d) ∑n f (e) = ∑ ∑n µ(d)f (e)
e| d
d|n
=
d
∑ ∑n µ(d)f (e) = ∑ f (e) ∑n µ(d)
e|n d|
=
d|n e|
e
e|n
d| e
∑ f (e)1e=n = f (n).
e|n
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let f be an arithmetic function and let
F(n) := ∑ f (d). If F is multiplicative, then so is f .
d|n
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let f be an arithmetic function and let
F(n) := ∑ f (d). If F is multiplicative, then so is f .
d|n
Proof.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let f be an arithmetic function and let
F(n) := ∑ f (d). If F is multiplicative, then so is f .
d|n
Proof. Let m, n ∈ N be so that (m, n) = 1.
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let f be an arithmetic function and let
F(n) := ∑ f (d). If F is multiplicative, then so is f .
d|n
Proof. Let m, n ∈ N be so that (m, n) = 1. Then
f (mn)
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let f be an arithmetic function and let
F(n) := ∑ f (d). If F is multiplicative, then so is f .
d|n
Proof. Let m, n ∈ N be so that (m, n) = 1. Then
mn f (mn) = ∑ µ(d)F
d
d|mn
Bernd Schröder
Multiplicative Functions
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let f be an arithmetic function and let
F(n) := ∑ f (d). If F is multiplicative, then so is f .
d|n
Proof. Let m, n ∈ N be so that (m, n) = 1. Then
mn m n
f (mn) = ∑ µ(d)F
= ∑ µ(dm dn )F
d
dm dn
d|mn
d |m,d |n
m
Bernd Schröder
Multiplicative Functions
n
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let f be an arithmetic function and let
F(n) := ∑ f (d). If F is multiplicative, then so is f .
d|n
Proof. Let m, n ∈ N be so that (m, n) = 1. Then
mn m n
f (mn) = ∑ µ(d)F
= ∑ µ(dm dn )F
d
dm dn
d|mn
dm |m,dn |n
m
n
= ∑ ∑ µ(dm )µ(dn )F
F
dm
dn
d |m d |n
m
Bernd Schröder
Multiplicative Functions
n
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let f be an arithmetic function and let
F(n) := ∑ f (d). If F is multiplicative, then so is f .
d|n
Proof. Let m, n ∈ N be so that (m, n) = 1. Then
mn m n
f (mn) = ∑ µ(d)F
= ∑ µ(dm dn )F
d
dm dn
d|mn
dm |m,dn |n
m
n
= ∑ ∑ µ(dm )µ(dn )F
F
dm
dn
dm |m dn |n
m
n
= ∑ µ(dm )F
µ(dn )F
∑
dm d |n
dn
d |m
m
Bernd Schröder
Multiplicative Functions
n
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let f be an arithmetic function and let
F(n) := ∑ f (d). If F is multiplicative, then so is f .
d|n
Proof. Let m, n ∈ N be so that (m, n) = 1. Then
mn m n
f (mn) = ∑ µ(d)F
= ∑ µ(dm dn )F
d
dm dn
d|mn
dm |m,dn |n
m
n
= ∑ ∑ µ(dm )µ(dn )F
F
dm
dn
dm |m dn |n
m
n
= ∑ µ(dm )F
µ(dn )F
= f (m)f (n)
∑
d
d
m
n
d |m
d |n
m
Bernd Schröder
Multiplicative Functions
n
Louisiana Tech University, College of Engineering and Science
Euler’s ϕ Function
Sum and Number of Divisors
Perfect Numbers and Mersenne Primes
Möbius Inversion
Theorem. Let f be an arithmetic function and let
F(n) := ∑ f (d). If F is multiplicative, then so is f .
d|n
Proof. Let m, n ∈ N be so that (m, n) = 1. Then
mn m n
f (mn) = ∑ µ(d)F
= ∑ µ(dm dn )F
d
dm dn
d|mn
dm |m,dn |n
m
n
= ∑ ∑ µ(dm )µ(dn )F
F
dm
dn
dm |m dn |n
m
n
= ∑ µ(dm )F
µ(dn )F
= f (m)f (n)
∑
d
d
m
n
d |m
d |n
m
Bernd Schröder
Multiplicative Functions
n
Louisiana Tech University, College of Engineering and Science