Problems on Chapter 1 17

(a)

a≤x

a∈A

L(a) = (1 + o(1))x as x → ∞,

(b) A(x) = (1 + o(1))

x

a1

dt

L(t)

as x → ∞.

Problem 1.12. Let L : [M, +∞) → R+, where M 0. Assume that

L ∈ C[M, +∞). Prove that

L ∈ L ⇐⇒

x

M

dt

L(t)

= (1 + o(1))

x

L(x)

as x → ∞.

Problem 1.13. Let A ⊂ N, with a1 being its smallest element. Let L :

[a1, +∞) → R+ be an increasing slowly oscillating function. Prove that

lim

x→∞

1

x

a≤x

a∈A

L(a) = 1 ⇐⇒ lim

x→∞

L(x)

x

a≤x

a∈A

1 = 1.

Problem 1.14. Let A = {p : p + 2 is prime}. It is conjectured that A(x) ∼

Cx/

log2

x (as x → ∞) for a certain positive constant C. Use the preceding

problem to show that this conjecture implies that

p≤x

p+2

prime

1

C

log2

p = (1 + o(1))x (x → ∞).

Problem 1.15. Prove that

1 ≤

n!

nne−n

√

2πn

≤

e1/12n

(n ≥ 1).