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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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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: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)
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)
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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