Lecture 8

Question 1

Give Dirichlet’s asyptotic formulas weaker version

Answer 1

∑_(n≤x) d(n) = x log x + O(x).

For all x ≥ 1

Question 2

Give Dirichlet’s asyptotic formula with a secondary main term

Improving the error rate

Answer 2

∑_(n≤x) d(n) = x log x + (2γ − 1)x + O(√x).

Question 3

What is the average order of the divisor function when alpha = 1

Answer 3

∑_(n≤x) σ(n) = 1/2 ζ(2)x^2 + O(x log x).

Question 4

What is the average order of the divisor function when alpha is not equal to 1

Answer 4

∑_(n≤x) σ_α(n) = ζ(α + 1)/(α + 1) x^(α+1) + O(x^β)

where β = max{1, α}.