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The sine sieve for primes ? - tommy1729 - 05/09/2015
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 RE: The sine sieve for primes ? - tommy1729 - 05/10/2015
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. Regards Tommy1729 |