4 1. PANORAMA OF ARITHMETIC FUNCTIONS
which is called the Riemann zeta-function. Actually ζ(s) was first introduced by
L. Euler who studied the distribution of prime numbers using the infinite product
formula
(1.5)
ζ(s) =
p
1 +
1
ps
+
1
p2s
+ · · ·
=
p
1 −
1
ps
−1
.
The series (1.4) and the product (1.5) converge absolutely in the half-plane s =
σ + it, σ 1. Since ζ(s) for s 1 is well approximated by the integral
∞
1
y−s
dy =
1
s − 1
as s → 1, it follows from (1.5) that
(1.6)
p
1
ps
∼ log
1
s − 1
, as s 1, s → 1.
By the Euler product for ζ(s) it follows that 1/ζ(s) also has a Dirichlet series
expansion
(1.7)
1
ζ(s)
=
p
1 −
1
ps
=
∞
1
μ(m)
ms
where μ(m) is the multiplicative function defined at prime powers by
(1.8) μ(p) = −1,
μ(pα)
= 0, if α 2.
This is a fascinating function (introduced by A. F. M¨ obius in 1832) which plays a
fundamental role in the theory of prime numbers.
Translating the obvious formula ζ(s) ·
ζ(s)−1
= 1 into the language of Dirichlet
convolution we obtain the δ-function
(1.9) δ(n) =
m|n
μ(m) =
1 if n = 1
0 if n = 1.
Clearly the two relations
(1.10) g = 1 ∗ f, f = μ ∗ g
are equivalent. This equivalence is called the M¨ obius inversion; more explicitly,
(1.11) g(n) =
d|n
f(d) ⇐⇒ f(n) =
d|n
μ(d)g(n/d).
If f, g are multiplicative, then so are f · g, f ∗ g. If g is multiplicative, then
(1.12)
d|n
μ(d)g(d) =
p|n
(
1 − g(p)
)
.
In various contexts one can view the left side of (1.12) as an “exclusion-inclusion”
procedure of events occurring at divisors d of n with densities g(d). Then the right
side of (1.12) represents the probability that none of the events associated with
prime divisors of n takes place.
