6 1. PANORAMA OF ARITHMETIC FUNCTIONS

A different class of arithmetic functions (very important for the study of prime

numbers) is obtained by differentiating relevant Dirichlet series

−Df (s) =

∞

1

f(n)(log

n)n−s.

In particular

−ζ (s) =

∞

1

(log

n)n−s.

Since L(n) = log n is additive, we have the formula

L · (f ∗ g) = (L · f) ∗ g + f ∗ (L · g)

which says that the multiplication by L is a derivation in the Dirichlet ring of

arithmetic functions.

By the Euler product we have

(1.14) log ζ(s) =

∞

l=1

p

l−1p−ls.

Hence differentiating we get

(1.15) −

ζ

ζ

(s) =

∞

1

Λ(n)n−s

with Λ(n) (popularly named von Mangoldt function) given by

(1.16) Λ(n) =

log p, if n = pα, α 1

0, otherwise.

From the left side of (1.15) we get

(1.17) Λ = μ ∗ L, Λ(n) =

d|n

μ(d) log

n

d

= −

d|n

μ(d) log d.

Hence, by M¨ obius inversion we get

(1.18) L = 1 ∗ Λ, log n =

d|n

Λ(d).

Similarly we define the von Mangoldt functions Λk of any degree k 0 by

(1.19) Λk = μ ∗

Lk,

Λk(n) =

d|n

μ(d) log

n

d

k

.

We have

(1.20)

Lk

= 1 ∗ Λk, (log

n)k

=

d|n

Λk(d).

Note that Λ0 = δ, Λ1 = Λ, and we have the recurrence formula

(1.21) Λk+1 = L · Λk + Λ ∗ Λk.

This follows by writing

Λk+1(n) =

d|n

μ(d) log

n

d

k

(log n − log d)

and using (1.18).