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.