Prime Numbers

1.4 Analytic number theory 35
(3) The properties of ζ in the “critical strip” 0 < Re(s) < 1 lead to deep
aspects of the distribution of primes, such as the essential error term in
the PNT.
On the point (1), we can prove Theorem 1.1.2 as follows:
Another proof of the infinitude of primes. We consider ζ(s) fors real, s>1.
Clearly, from relation (1.17), ζ(s) diverges as s → 1 + because the harmonic
sum 1/n is divergent. Indeed, for s>1,
ζ(s) >
n≤1/(s−1)
≥ e −1/e
n −s =
n≤1/(s−1)
n≤1/(s−1)
n −1 n −(s−1)
n −1 >e −1/e | ln(s − 1)|.
But if there were only finitely many primes, the product in (1.18) would tend
to a finite limit as s → 1 + , a contradiction. ✷
The above proof actually can be used to show that the sum of the
reciprocals of the primes diverges. Indeed,
⎛
ln ⎝
(1 − p −s ) −1
⎞
⎠ = −
ln(1 − p −s )=
p −s + O(1), (1.19)
p∈P
p∈P
uniformly for s>1. Since the left side of (1.19) goes to ∞ as s → 1 + and
since p −s < p −1 when s > 1, the sum
p∈P p−1 is divergent. (Compare
with Exercise 1.20.) It is by a similar device that Dirichlet was able to prove
Theorem 1.1.5; see Section 1.4.3.
Incidentally, one can derive much more concerning the partial sums of 1/p
(henceforth we suppress the notation p ∈P, understanding that the index p
is to be a prime variable unless otherwise specified):
Theorem 1.4.2 (Mertens). As x →∞,
p≤x
1 − 1
p
p∈P
∼ e−γ
, (1.20)
ln x
where γ is the Euler constant. Taking the logarithm of this relation, we have
p≤x
for the Mertens constant B defined as
B = γ +
1
=lnlnx + B + o(1), (1.21)
p
p
ln
1 − 1
p
+ 1
.
p