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.