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21439Re: product convergence

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  • djbroadhurst
    May 4, 2010
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      --- In primenumbers@yahoogroups.com,
      "djbroadhurst" <d.broadhurst@...> wrote:

      > prod(2<p<x, (p-2)/p) = O(1/log(x)^2)

      So let's work out the constant, say K, in

      prod(2<p<x, (p-2)/p) ~ K/log(x)^2

      We should use the twin-prime constant

      C2 = prod (2<p, p*(p-2)/(p-1)^2) = 0.6601618158...

      and then use the square of Mertens' formula,
      remembering that the latter includes p = 2.

      K = C2*(exp(-Euler)*2)^2 =
      0.832429065661945278030805943531465575045445318077417053240894...

      Sanity check:

      default(primelimit,10^8);
      \p5
      P=1.;x=10^8;forprime(p=3,x,P*=1-2/p);print(P*log(x)^2);

      0.83242

      Looks OK to me...

      David
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