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).
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