Theorem 2.1
(Merten’s Theorem)
.
Assuming that:
x
≥
3
Then
(i)
∑
p
≤
x
log
p
p
=
log
x
+
O
(
1
)
(ii)
∑
p
≤
x
1
p
=
log
log
x
+
M
+
O
(
1
log
x
)
(for some
M
∈
ℝ
)
(iii)
∏
p
≤
x
(
1
−
1
p
)
=
c
+
o
(
1
)
log
x
(for some
c
>
0
)