14 1. Arithmetic Functions
yields
n≤x
Λ(n)
n
=
T(x)
x
+ O
ψ(x)
x
= log(x) + O(1)
by Stirling’s formula and Proposition 1.1. The series
p
∞
k=2
log(p)
pk
converges, and Λ(n) is zero off the prime powers.
Sometimes it is necessary to remove a factor from the terms of a sum.
This is an important application of the partial summation formula, and is
illustrated in the proof of the next result.
Proposition 1.10. The estimate
p≤x
1
p
= log log(x) + a + O
1
log(x)
holds with some constant a.
Proof. First
p≤x
1
p
=
p≤x
log(p)
p
1
log(p)
=
⎛
⎝
p≤x
log(p)⎠
p
⎞
1
log(x)
−
x
2
⎛
⎝
p≤u
log(p)⎠
p
⎞
(−1) du
u
log2(u)
by partial summation. Then
p≤x
1
p
= 1 + O
1
log(x)
+
x
2
du
u log(u)
+
x
2
⎛
⎝
p≤u
log(p)
p
−
log(u)⎠
⎞
du
u
log2(u)
= 1 + O
1
log(x)
+ log log(x) − log log(2)
+
∞
2
⎛
⎝
p≤u
log(p)
p
−
log(u)⎠
⎞
du
u
log2(u)
+
∞
x
O(1)
u
log2(u)
du
= log log(x) + a + O
1
log(x)
by Proposition 1.9 and integration by parts.
