Lecture 4

Question 1

Define the Dirichlet product/Dirichlet convolution

Answer 1

Given two arithemetical functions f and g, we define their Dirichlet Product (or Dirichlet Convolution) to be the arithmetical function h defined by the equation

• h(n) = ∑_(d∣n) f(d)g (n/d) .

Question 2

What is the function N

Answer 2

N(n) = n for all n.

Question 3

What arithmetical properties hold for the Dirchlet convolution

Answer 3

Communicative and associative

Question 4

Define the Dirichlet inverse

Answer 4

If f is an arithmetical function with f(1) ≠ 0, then there exists a unique arithemetical function f^(−1) such that:

• f ∗ f^(−1) = f^(−1) ∗ f = I.

Question 5

Define the unit function

Answer 5

• u(n) = 1 for all n.

Question 6

Give the Mobius inversion formula

Answer 6

• f(n) = ∑_(d∣n) g(d)

implies

• g(n) = ∑_(d∣n) f(d)µ(n/d) .
• And vice versa

Question 7

What is the Von-Mangoldt Function

Answer 7

For any integer n ≥ 1, we define the VonMangoldt function Λ(n) by:

• Λ(n) = {log p if n = p^m for some prime p and some m ≥ 1, 0 otherwise}

Question 8

What is the sum of the Von-Mangoldt function over divisors

Answer 8

• log n = ∑_(d∣n) Λ(d).

Question 9

How can we equate the Von-Mangoldt function with the Mobius function

Answer 9

Λ(n) = ∑(d∣n) µ(d) log (n/d) = −∑(d∣n) µ(d) log d