## Transcribed Text

Exercise 2.1.4. Complete the proof of Theorem 2.1.3(c) as follows:
(a) For ∈ (0, 1] and x ≥ 2, show that
X
p≤x
log
1 −
1
p
=
X
p≤x
log
1 −
1
p
1+/ log x
+ O().
(b) For s ∈ (1, 2], show that
X
p
log
1 −
1
p
s
= log(s − 1) + O(|s − 1|).
(c) For ∈ (0, 1] and x ≥ 2, show that
X
p>x
log
1 −
1
p
1+/ log x
= −
Z ∞
e
−u
u
du + O
1
log x
.
[Hint: Theorem 2.1.3(a).] Conclude that
X
p≤x
log
1 −
1
p
= − log log x +
Z ∞
0
e
−u − 1[0,1](u)
u
du + O
+
1
log x
,
where 1A denotes the characteristic function of the set A.
(d) Note that
γ = lim
N→∞
− log N +
Z 1
0
(1 + x + · · · + x
N−1
)dx
.
Make the change of variables x = 1 − u/N and deduce that γ =
R ∞
0
1[0,1](u)−e−u
u
du to
complete the proof of Theorem 2.1.3(c).
Exercise 2.2.5. Given x, y ≥ 1, we define
Φ(x, y) = #{n ≤ x : p|n ⇒ p > y}.
Prove that there is a constant c ≥ 2 such that
Φ(x, y)
x
log y
(2 ≤ y ≤ x/c)
Exercise 2.3.5. (a) Show that
X
n≤x
log(x/n) x (x ≥ 2).
(b) Prove that µ log = −µ ∗ Λ and deduce that
M(x) log x = −
X
d≤x
Λ(d)M(x/d) + O(x).
(c) Show that
lim inf
x→∞
M(x)
x
+ lim sup
x→∞
M(x)
x
= 0.

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