Lecture 25 (PNT lemmas)

Question 1

For any arithmetical function a(n), let A(x) = ∑n≤x a(n) where A(x) = 0 if x < 1. Then find

• ∑_(n≤x) (x − n)a(n)

Answer 1

For any arithmetical function a(n), let A(x) = ∑n≤x a(n) where A(x) = 0 if x < 1. Then

• ∑_(n≤x) (x − n)a(n) = ∫^x_1 A(t) dt

Question 2

Let A(x) = ∑n≤x a(n) and let A1(x) = ∫^x_1 A(t) dt. Assume a(n) ≥ 0 for all n. If we have the asymptotic formula

• A1(x) ∼ Lx^c as x → ∞

for some c > 0 and L > 0, what can we say about A(x)

Answer 2

Let A(x) = ∑n≤x a(n) and let A1(x) = ∫^x_1 A(t) dt. Assume a(n) ≥ 0 for all n. If we have the asymptotic formula

• A1(x) ∼ Lx^c as x → ∞

for some c > 0 and L > 0, then we have

• A(x) ∼ cLx^(c−1) as x → ∞.

Question 2

Let A(x) = ∑n≤x a(n) and let A1(x) = ∫^x_1 A(t) dt. Assume a(n) ≥ 0 for all n. If we have the asymptotic formula

• A1(x) ∼ Lx^c as x → ∞

for some c > 0 and L > 0, what can we say about A(x)

Answer 2

Let A(x) = ∑n≤x a(n) and let A1(x) = ∫^x_1 A(t) dt. Assume a(n) ≥ 0 for all n. If we have the asymptotic formula

• A1(x) ∼ Lx^c as x → ∞

for some c > 0 and L > 0, then we have

• A(x) ∼ cLx^(c−1) as x → ∞.

Question 3

We have

• ψ1(x) = ∑_(n≤x) (x − n)Λ(n)

what does the asymptotic formula imply:

• ψ1(x) ∼1/2 x^2 as x → ∞

Answer 3

ψ(x) ∼ x as x → ∞

Question 4

If c > 0 and u > 0, then for every integer k ≥ 1, find
* 1/(2πi) ∫^(c+i∞)_(c−i∞) u^(−z)/(z(z + 1) . . . (z + k)) dz

Answer 4

If c > 0 and u > 0, then for every integer k ≥ 1, we have

• 1/(2πi) ∫^(c+i∞)_(c−i∞) u^(−z)/(z(z + 1) . . . (z + k)) dz =
  
  • 1/k! (1 − u)^k if 0 < u ≤ 1,
  • 0 if u > 1,

the integral being absolutely convergent.