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).
