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The sine sieve for primes ?
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 ?


Maybe the fact that prod (sin^2(n) - 25/16) stays bounded and diverges has something to do with this.

This suggests that Maybe we should replace s with similar to make things nicer.
Im also unsure about that integral ... Maybe the values of F are too close to 0 most of the time.

I think with strong hesitation that for large n with respect to x ,
The function F Goes to exp(-2n ln(n) + 2n) prod gamma(p_n)^(2/p_n) asymptotically with measure 1. This estimate is based on statistical thinking and prod p_n ~ exp(n ln n - n).

This estimate hints at correcting factors for the integrand also known as weights.

Atthough all this suggests we are not talking of the simplest or best sieve type function , it is intuïtive and therefore intresting.

A lot of work to do but lots of inspiration.
The way i like math usually.



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