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. So if for √ x ≤ y n ≤ x the events p|n and q|n for distinct primes p, q ≤ √ 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.

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