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.