Prime Numbers

34 Chapter 1 PRIMES!
have identities such as
ζ(s) = s
∞
− s (x −⌊x⌋)x
s − 1 −s−1 dx.
1
But this formula continues to apply in the region Re(s) > 0, s = 1,sowe
may take this integral representation as the definition of ζ(s) for the extended
region. The equation also shows the claimed nature of the singularity at s =1,
and other phenomena, such as the fact that ζ has no zeros on the positive
real axis. There are yet other analytic representations that give continuation
to all complex values of s.
The connection with prime numbers was noticed earlier by Euler (with
the variable s real), in the form of a beautiful relation that can be thought of
as an analytic version of the fundamental Theorem 1.1.1:
Theorem 1.4.1 (Euler). For Re(s) > 1 and P the set of primes,
ζ(s) =
p∈P
(1 − p −s ) −1 . (1.18)
Proof. The “Euler factor” (1 − p −s ) −1 mayberewrittenasthesumofa
geometric progression: 1 + p −s + p −2s + ···. We consider the operation of
multiplying together all of these separate progressions. The general term in the
multiplied-out result will be
p∈P p−aps ,whereeachap is a positive integer
or 0, and all but finitely many of these ap are 0. Thus the general term is n −s
for some natural number n, and by Theorem 1.1.1, each such n occurs once
and only once. Thus the right side of the equation is equal to the left side of
the equation, which completes the proof. ✷
As was known to Euler, the zeta function admits various closed-form
evaluations, such as
ζ(2) = π 2 /6,
ζ(4) = π 4 /90,
and in general, ζ(n) for even n is known; although not a single ζ(n) for odd
n > 2 is known in closed form. But the real power of the Riemann zeta
function, in regard to prime number studies, lies in the function’s properties
for Re(s) ≤ 1. Closed-form evaluations such as
ζ(0) = −1/2
are sometimes possible in this region. Here are some salient facts about
theoretical applications of ζ:
(1) The fact that ζ(s) →∞as s → 1 implies the infinitude of primes.
(2) The fact that ζ(s) has no zeros on the line Re(s) = 1 leads to the prime
number Theorem 1.1.4.