Re: Simple but non strngent estomation of the prime-counting function.

On May 20, 7:42 am, Gerry Myerson <myer...@xxxxxxxxxxxxxxxxxxx> wrote:
Simple estimation of the number of primes <=n.

The probability that n is prime is:

P(n) ~=  (1-1/2).(1-1/3).(1-1/5)...........(1-1/sqrt(n))

<snip!>

Can this sort of reasoning be made stringent?

One difficulty is that even if you get the best possible estimate for the
product above, it leads to an estimate for the number of primes that is
wrong by a constant factor. The reason is that the primes are not
independent.

A slight rephrase. It isn't that the 'primes are not independent',
but rather once the product 2*3*5*.... exceeds n, the probability
that
another prime divides n is no longer independent of the probability
that
one or more of 2,3,5,7.... divides n. In other words they can't ALL
divide
n, so the probability that one divides n isn't independent of the
probability that
another divides n.

Mertens was the first to come up against this problem,
if you search for that name you will probably find what you want.

Hardy & Wright, Intro to the Theory of Numbers has a nice exposition.
.

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