4. The Euler-Riemann zeta function 5

(The condition that (s) 1 guarantees convergence of the series.) In the

analytic approach to prime number theory, this function occupies a central

position. Because of this text’s emphasis on elementary methods, the zeta

function will not play a large role for us, but it should be stressed that in

many of the deeper investigations into the distribution of primes, the zeta

function is an indispensable tool.

Riemann introduced the study of ζ(s) as a function of a complex vari-

able in an 1859 memoir on the distribution of primes [Rie59]. But the

connection between the zeta function and prime number theory goes back

earlier. Over a hundred years prior to Riemann’s study, Euler had looked

at the same series for real s and had shown that [Eul37, Theorema 8]

(1.2)

∞

n=1

1

ns

=

p

1

1 −

1

ps

(s 1).

This is often called an analytic statement of unique factorization. To see

why, notice that formally (i.e., disregarding matters of convergence)

p

1 +

1

ps

+

1

p2s

+ · · · =

∞

n=1

an

ns

,

where an counts the number of factorizations of n into prime powers. Thus

unique factorization, the statement that an = 1 for all n, is equivalent to

the statement that (1.2) holds as a formal product of Dirichlet

series.1

This,

in turn, is equivalent to the validity of (1.2) for all real s 1 (or even a

sequence of s tending to ∞) by a standard result in the theory of Dirichlet

series (see, e.g., [Apo76, Theorem 11.3]).

Euler’s product expansion of the zeta function is the first example of

what is now called an Euler factorization. We now prove (following [Hua82])

a theorem giving general conditions for the validity of such factorizations.

Theorem 1.2 (Euler factorizations). Let f be a multiplicative function.

Then

(1.3)

∞

n=1

f(n) =

p

(

1 + f(p) +

f(p2)

+ · · ·

)

if either of the following two conditions holds:

(i)

∑∞

n=1

|f(n)| converges.

(ii)

p

(

1 + |f(p)| +

|f(p2)|

+ · · ·

)

converges.

1Here

a Dirichlet series is a series of the form F (s) =

∑∞

n=1

cn/ns,

where each cn is a

complex number.