Lecture 9

Question 1

Find ∑(n≤x) φ(n)

For x > 1

Answer 1

3/(π^2) x^2 + O(x log x).

Question 2

What’s the order of µ(n) and Λ(n)

Answer 2

• lim_(x→∞)1/x ∑_(n≤x) µ(n) = 0
• lim_(x→∞) 1/x ∑_(n≤x) Λ(n) = 1

Question 3

State the partial sum of the covolution h = f*g

Answer 3

let H(x) = ∑(n≤x) h(n), F(x) = ∑(n≤x) f(n) and G(x) = ∑_(n≤x) g(n).

Then H(x) = ∑(n≤x) f(n)G(x/n) = ∑(n≤x) g(n) F(x/n) .

Question 4

Give the partial sum of µ(n)

Answer 4

∑(n≤x) µ(n) [x/n] = ∑(n≤x)∑(d∣n) µ(d) = ∑(n≤x) [1/n] = 1

Question 5

Give the partial sum of Λ(n)

Answer 5

∑(n≤x) Λ(n) [x/n] = ∑(n≤x)∑(d∣n) Λ(d) = ∑(n≤x) log n = log[x]!.