The sine sieve for primes ?

[tommy1729]

Let s(x) = sin^2 (x/pi).
Let the nth prime be p_n.
Mick and myself are considering the prime Sieve approximation :

F(x,n) = s(x/2)s(x/3)...s(x/p_n)

F(x,n)/F(1,n) = g(x,n)

H(x,n) = integral g(x,n) dx

Many questions occur.
How good is h compared to the prime counting function ?
How does F(1,n) grow ?

Is there much difference between taking n such that p_n ~ sqrt x and taking
p_n ~ x ?

Regards

Tommy1729