Elementary proof of PNT

Question 1

Selberg’s Asymptotic Formula

Answer 1

For x > 0, we have
* ψ(x) log x + ∑_(n≤x) Λ(n)ψ (x/n) = 2x log x + O(x)

Question 2

What are the steps for the elementary proof of the PNT

Answer 2

See lecture notes page 63

Question 3

In the elementary proof of the Prime Number Theorem

what do we set σ(x) equal to

Answer 3

e^(−x)ψ(e^x) − 1

Question 4

In the elementary proof of the Prime Number Theorem

What does Selberg’s identity imply

Answer 4

|σ(x)∣x^2 ≤ 2∫^x_0 ∫^y_0 ∣σ(u)∣dudy + O(x)

Question 5

In the elementary proof of the Prime Number Theorem

What is the prime number theorem equivalent to showing

Answer 5

σ(x) → 0 as x → ∞.

Question 6

In the elementary proof of the Prime Number Theorem

If we let C = lim sup_(x→∞) ∣σ(x)∣

What is the PNT equivalent to showing

Answer 6

C=0

Question 7

In the elementary proof of the Prime Number Theorem

What do we assume about C

Answer 7

Assume C>0 to gain a contradiction