I have been wondering whether there is an easy way to rationalize the density of the prime numbers. The density D(N) is about 1/ln(N).

From the Sieve of Eratosthenes we can approximate the density as

D(N) ≅ (1−1/p₁) ... (1−1/p_i), where p₁ ... p_i are the primes less than √N.

The sieve actually does better than random; the efficiency increases with N and appears to be asymptotic to about 12% better than random (it's easy to see why it does better than random). But since (1−1/p₁)...(1−1/p_i) / D(N) does appear to converge on a limit, we can use this estimate to construct a functional equation for the density of the prime numbers.

If we approximate the density by a continuous function, we get

D(x + dx) = D(x) (1−1/√x)^{D(√x) d(√x)}.

Bringing the exponent down with a Taylor series expansion, this becomes

D(x) (1 + ln (1−1/√x) D(√x) dx)
                        2 √x

for x large this approaches

D(x) (1 − D(√x) dx)
                2 x

giving the functional equation

D'(x) = − D(√x) D(x)
                 2 x

The answer, D(x) = 1 / ln(x), is a solution to this equation.

Are there other solutions??

back to math page