--- In

primenumbers@yahoogroups.com,

<chitatel2000@...> wrote:

> Number of primes in the interval [a ^ 2, b ^ 2]. b-a = 1

The number Q of such primes is asymptotic to a/log(a).

Sergey over-estimates Q by the well-known percentage

2*exp(-Euler) - 1 =~ 12.29%

from bad reasoning about the sieve of Eratosthenes.

Example: Set a = 10^6.

There are 37,607,912,018 primes less than 10^12.

There are 37,607,984,431 primes less than 10^12 + 2*10^6 +1.

Here, Q = 72413 is indeed close to 10^6/log(10^6) =~ 72382.

Now look at Sergey's false estimate:

(2*10^6 + 1)*prod(p=2,10^6,1-if(isprime(p),1./p)) =~ 81276.

This leaves him with the egregious error

E = 81276 - 72413 = 8863, i.e. 12.24% of Q, which is indeed

close to the Mertens estimate, 12.29% of Q, provided above.

David