Chapter 2 definitions

Question 1

coprime

Answer 1

Two integers n and m are coprime in ℤ if their only common factors are 1 and -1

Question 2

multiplicative

Answer 2

The function f: ℕ -> ℂ is multiplicative if f(nm)=f(n)f(m) where n and m are coprime

Question 3

τ(n)

Answer 3

the number of positive factors of n

Question 4

σ(n)

Answer 4

the sum of the positive factors of n

Question 5

Φ(n)

Answer 5

The number of numbers in the range 1,..,n that are coprime to n

Question 6

f̂

Answer 6

Let f:ℕ -> ℂ be a function and let n∈ℕ. then
Σ d|n f(d) is defined as
f(d₁) + f(d₂) + … + f(dₖ)

Question 7

Mobius function

Answer 7

The mobius function μ: ℕ -> ℂ is given by

μ(n) = {1 if n=1, 0 if p²|n for some prime p, (-1)ᵏ if n = p₁p₂…pₙ with all distinct primes

Question 8

Euler Phi function

Answer 8

We define the Euler function Φ: ℕ -> ℂ by

Φ(n) = the number of numbers in the range 1,…,n that are coprime to n