2 Dirichlet convolution To better understand the Möbius inversion formula, we introduce the following concept. Definition 2.1. Given two arithmetic functions f and g, their Dirichlet convolution, denoted as fg, is defined as follows: (f*g) (n) = [ f(d)g (7) d|n where the summation is taken over the divisor set (n) := {d|d is a divisor of n}. The basic property of Dirichlet convolution is the following. Theorem 2.2. The set of arithmetic functions, together with the Dirichlet convolution, forms a commutative monoid. Namely, the followings hold. 1. The binary operation is associative: for any arithmetic functions f, g, and h, (f*g) *h =f*(g*h). (2.2) 2. The binary operation has a neutral element. Indeed, let & be the function defined as follows: 8(n) := {} Then & is a neutral element for the binary operation ★: for any arithmetic function f, 8* f = f*8 = f. (2.3) (2.1) if n = 1, if otherwise. 3. The binary operation is commutative: for any arithmetic functions f and g, f*g=g*f. Proof. (PROOF NEED TO BE FILLED) (2.4) ☐
2 Dirichlet convolution To better understand the Möbius inversion formula, we introduce the following concept. Definition 2.1. Given two arithmetic functions f and g, their Dirichlet convolution, denoted as fg, is defined as follows: (f*g) (n) = [ f(d)g (7) d|n where the summation is taken over the divisor set (n) := {d|d is a divisor of n}. The basic property of Dirichlet convolution is the following. Theorem 2.2. The set of arithmetic functions, together with the Dirichlet convolution, forms a commutative monoid. Namely, the followings hold. 1. The binary operation is associative: for any arithmetic functions f, g, and h, (f*g) *h =f*(g*h). (2.2) 2. The binary operation has a neutral element. Indeed, let & be the function defined as follows: 8(n) := {} Then & is a neutral element for the binary operation ★: for any arithmetic function f, 8* f = f*8 = f. (2.3) (2.1) if n = 1, if otherwise. 3. The binary operation is commutative: for any arithmetic functions f and g, f*g=g*f. Proof. (PROOF NEED TO BE FILLED) (2.4) ☐
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Fill out the blank where it says (PROOF NEED TO BE FILLED)
Transcribed Image Text:2 Dirichlet convolution
To better understand the Möbius inversion formula, we introduce the following concept.
Definition 2.1. Given two arithmetic functions f and g, their Dirichlet convolution, denoted
as fg, is defined as follows:
(f*g)(n) = [ f(d)g (7)
d|n
where the summation is taken over the divisor set D(n) := {d|d is a divisor of n}.
The basic property of Dirichlet convolution is the following.
Theorem 2.2. The set of arithmetic functions, together with the Dirichlet convolution,
forms a commutative monoid. Namely, the followings hold.
1. The binary operation is associative: for any arithmetic functions f, g, and h,
(f*g) *h = f*(g*h).
(2.2)
2. The binary operation has a neutral element. Indeed, let & be the function defined as
follows:
8(n) :=
{
Then & is a neutral element for the binary operation *: for any arithmetic function f,
d* f = f*8 = f.
(2.3)
(2.1)
if n = 1,
if otherwise.
3. The binary operation is commutative: for any arithmetic functions f and g,
f*g=g* f.
Proof (PROOF NEED TO BE FILLED)
(2.4)
Transcribed Image Text:1 The Möbius function and the inversion formula
To state the Möbius inversion formula, we need the following notions.
Definition 1.1. We say that a positive integer n is square-free if n is not divisible by p² for
any prime number p. The Möbius function is defined as follows:
μ(η) :=
1
£₁
0
if n = 1,
if n is not sqaure-free,
if n is sqaure-free and has exactly t prime divisors.
The Möbius function is an arithmetic function. Namely, it is a complex-valued function
fl
defined on the set Z+ of positive integers.
Definition 1.2. The Möbius transformation of an arithmetic function f is the function f
defined by the formula
(1.1)
f(n) := Σf(d).
d|n
Then the Möbius inversion formula can be stated as follows.
Theorem 1.3 (Möbius inversion formula). For any arithmetic function f, we have
f(n) = Σμ(4)f(a).
d|n
1
(1.2)
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