I have been wondering whether there is an easy way to rationalize the
density of the prime numbers. The density

From the Sieve of Eratosthenes we can approximate the density as

_{1}) ...
(1−1/p_{i}),_{1} ... p_{i}

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})...(1−1/p_{i}) / D(N)

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

^{D(√x)
d(√x)}.

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

__ln (1−1/√x) D(√x) dx__)

for x large this approaches

__D(√x) dx__)

giving the functional equation

__ D(√x) D(x) __

The answer,

Are there other solutions??

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