1.4. The Mertens estimates 15
The next result is of considerable significance in prime number theory for
various considerations of a probabilistic nature. The fact that the constant
b in the formula is positive is important in such contexts. Actually b equals
the Euler-Mascheroni constant γ, though we won’t prove this.
Proposition 1.11 (Mertens’ formula). The estimate
p≤x
1 −
1
p
∼
e−b
log(x)
holds with some constant b.
Proof. First
log
⎛
⎝
p≤x
1 −
1
p
⎞
⎠
=
p≤x
log 1 −
1
p
= −
p≤x
∞
k=1
1
kpk
where
−
p≤x
∞
k=1
1
kpk
= −
p≤x
1
p
−
p
∞
k=2
1
kpk
+
px
∞
k=2
1
kpk
.
The first term on the right-hand side may be estimated by means of Propo-
sition 1.10, the second term is a convergent infinite series, and the third
term tends to zero as x → +∞. Thus
p≤x
1 −
1
p
∼ exp(− log log(x) − b) =
e−b
log(x)
by exponentiating.
About half of all the integers n with y n ≤ x for x and x − y large
are even, one third are divisible by three, and so forth. A suggestive way
of phrasing this observation is to say that the chance of a randomly chosen
large integer n being divisible by a prime p is 1/p. An integer n ≥ 2 that is
not divisible by any prime p ≤
√
n is itself prime. if for
√
x ≤ y n ≤ x
the events p|n and q|n for distinct primes p, q ≤
√So
x are independent in the
sense of probability theory, the chance of n being prime should be
p≤
√
x
1 −
1
p
∼
e−γ
log(
√
x)
=
2e−γ
log(x)
1.12
log(x)
.
But the density of the primes near x is close to 1/ log(x) by the Prime Num-
ber Theorem. We conclude that the events p|n and q|n are not independent.
It is easy to persuade oneself that independence must hold for pairs of dis-
tinct primes that are very small compared with n. Thus Mertens’ formula
reveals an aspect of divisibility of integers by comparatively large primes.
