18 1. Arithmetic Functions
us, for example, calculate the Dirichlet convolution 1 ∗ φ. Both factors are
multiplicative, so the calculation
(1 ∗
φ)(pα)
=
pβ|pα
1 ·
φ(pβ)
= 1 +
α
β=1
(p −
1)pβ−1
=
pα
yields the convolution identity 1 ∗ φ = id of Gauss.
The M¨ obius mu-function μ is the unique multiplicative arithmetic func-
tion with values μ(p) = −1 on the primes p, and values
μ(pk)
= 0 on the
prime powers
pk
with k ≥ 2. The M¨ obius function has a strong combinato-
rial flavor. It is closely connected to the principle of inclusion and exclusion,
and to the fact that the integers form a partially ordered set under the re-
lation of divisibility. The importance of the M¨ obius function is due to the
convolution identity
d|n
μ(d) = e(n).
There are many ways to establish that 1 ∗ μ = e, but the quickest is to
recall that μ is multiplicative. Then so is 1 ∗ μ, and thus (1 ∗
μ)(pα)
=
1 + (−1) + 0 + 0 + · · · = 0 yields (1 ∗ μ)(n) = 0 for all n ≥ 2.
Proposition 1.13 (First M¨ obius inversion formula). If g = 1 ∗ f then
f = μ ∗ g and conversely.
Proof. If g = 1 ∗ f, then μ ∗ g = μ ∗ (1 ∗ f) = (μ ∗ 1) ∗ f = e ∗ f = f, and if
f = μ ∗ g then 1 ∗ f = 1 ∗ (μ ∗ g) = (1 ∗ μ) ∗ g = e ∗ g = g.
This shows that 1 is a unit in the Dirichlet ring, and μ is its multiplicative
inverse. An arithmetic function f is a unit if and only if f(1) = 0. Under
this condition a Dirichlet inverse g for f may be constructed incrementally
from
g(n) = e(n) −
n=d|n
f
n
d
g(d).
The relation f(1)g(1) = e(1) = 1 shows that the condition f(1) = 0 is
necessary. Constructing an explicit Dirichlet inverse is usually infeasible,
except in the very important case when f is multiplicative. The first M¨obius
inversion formula is often formulated as the statement
“f(n) =
d|n
μ(d)g
n
d
if and only if g(n) =
d|n
f(d)”
about divisor sums.
