i proved that:

pi(n) - pi(âˆšn) = n Ã— (1/2 Ã— 2/3 Ã— ... Ã— p-1/p)

p = largest prime less than or equal to âˆšn

prime number theorem: pi(n) = n/ln(n)

lim pi(âˆšn) / pi(n) = lim âˆšn/ln(âˆšn) / n/ln(n) = lim 2/âˆšn = 0

=> pi(n) - piâˆšn ~ pi(n) = n Ã— (1/2 Ã— 2/3 Ã— ... Ã— p-1/p)

p < âˆšn

and

pi(n^2) = n^2 Ã— (1/2 Ã— 2/3 Ã— ... Ã— q-1/q)

q < n

=> pi(n^2)/pi(n) = n Ã— ( p-1/p Ã— ... Ã— q-1/q)

p > âˆšn , q < n

and

pi(n^2)/pi(n) = n^2/ln(n^2) / n/ln(n) = n/2

=> n/2 = n Ã— (p-1/p Ã— ... Ã— q-1/q)

=> lim p-1/p Ã— ... Ã— q-1/q = 1/2

p>âˆšn , q<n

for n=100 it is 0.526 ~ 1/2

for n=1000 it is 0.513 ~ 1/2