--- In

primenumbers@yahoogroups.com,

"djbroadhurst" <d.broadhurst@...> wrote:

> I have already explained why your error is unbounded

Off-list, Sergey asked me for an example of a large error.

Let p[n] = 10^1000 + 453.

Then p[n-1] = 10^1000 - 1769 is the previous prime.

Sergey defines Q as the number of primes

between p[n-1]^2 and p[n]^2.

Using the prime number theorem, we may estimate

Q =~ (p[n]^2 - p[n-1]^2)/log(p[n]^2) =~ 9.65*10^999

Sergey defines his "error" E by

E = (p[n]^2 - p[n-1]^2)*prod(k=1,n,1-1/p[k]) - Q

Using Mertens' theorem

http://mathworld.wolfram.com/MertensTheorem.html
we may easily estimate

E =~ 2*exp(-Euler)*Q - Q =~ 1.19*10^999

whose integer part has 1000 decimal digits.

Clearly the error is unbounded, since it is

of the same order as the n-th prime, p[n].

I remark that Sergei has already advertised his "blogspot" in 9

messages to this list, while making no significant contribution

to our own discussions. I hope that we shall not receive a 10th

advertisement for a site that seems to make no attempt to learn

from careful correction of its obvious failures.

David Broadhurst