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).
