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.