## 21440product convergence

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• May 5, 2010
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> Messages in this topic (5)
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> 1b. Re: product convergence
> Date: Tue May 4, 2010 1:17 pm ((PDT))
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>> prod(2<p<x, (p-2)/p) = O(1/log(x)^2)
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> So let's work out the constant, say K, in
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> prod(2<p<x, (p-2)/p) ~ K/log(x)^2
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> We should use the twin-prime constant
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> C2 = prod (2<p, p*(p-2)/(p-1)^2) = 0.6601618158...
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> and then use the square of Mertens' formula,
> remembering that the latter includes p = 2.
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> K = C2*(exp(-Euler)*2)^2 =
> 0.832429065661945278030805943531465575045445318077417053240894...
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> Sanity check:
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> default(primelimit,10^8);
> \p5
> P=1.;x=10^8;forprime(p=3,x,P*=1-2/p);print(P*log(x)^2);
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> 0.83242
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> Looks OK to me...
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> David
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Hmm.... And I thought I had proven that the product converged to zero.

David, what is wrong with my "Proof" below which I had already sent to Tim?

Kermit

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> Hello Tim.
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> http://en.wikipedia.org/wiki/Infinite_product
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> The infinite product 3/5 5/7 9/11 ...
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> converges to zero.
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> would converge to a positive number between 0 and 1 only if
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> the infinite product
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> 5/3 7/5 11/9 ..... converged to a positive number > 1.
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> 5/3 7/5 11/9 ..... = (1 + 2/3) (1 + 2/5) (1 + 2/9) (1 + 2/11) (1 +
> 2/15) (1 + 2/17) ...
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> 1 + 2/3 + 2/5 + 2/9 + 2 / 11 + .... =< (1 + 2/3) (1 + 2/5) (1 + 2/9)
> (1 + 2/11) (1 + 2/15) (1 + 2/17) ... =< exp( 2/3 + 2/5 + 2/9 + 2/11
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