Lectures 12 & 13

Question 1

Use Abel’s identity to express θ(x) as an integral

Answer 1

θ(x) = π(x) log x − ∫^x_2 π(t)/t dt

Question 2

Use Abel’s identity to express π(x) as an integral

Answer 2

π(x) = θ(x)/log x + ∫^x_2 θ(t)/tlog^2(t) dt.

Question 3

State the prime number theorem as the asymptotic value of the nth prime

Answer 3

lim_(n→∞) p_n/(nlog n) = 1.

Question 4

What are the bounds for π(n)

Answer 4

1/6(n/log n) < π(n) < 6(n/log n)
.

Question 5

Give the bounds for the nth prime

Answer 5

1/6 nlog n < pn < 12 (nlog n + nlog 12/e)

Question 6

Shapiro’s Tauberian theorem

Answer 6

Let {a(n)} be a non-negative sequence such that
* ∑_(n≤x) a(n) [x/n] = x log x + O(x)
for all x ≥ 1. Then

1. For x ≥ 1, we have
   
   • ∑_(n≤x) a(n)/n = log x + O(1)
2. There exists a constant B > 0 such that
   
   • ∑_(n≤x) a(n) ≤ Bx, for all x ≥ 1.
3. There exists a constant A > 0 and an x0 > 0 such that
   
   • ∑_(n≤x) a(n) ≥ Ax, for all x ≥ x0