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• From: Werner D. Sand Date: 12/04/05 04:58:46 To: primenumbers@yahoogroups.com Subject: [PrimeNumbers] Re: product I suspect the product 2/3 * 7/5 * 11/13 *
Message 1 of 11 , Dec 4 3:30 PM
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From: Werner D. Sand
Date: 12/04/05 04:58:46

I suspect the product 2/3 * 7/5 * 11/13 * 19/17 * ... to converge
to 1, oscillating around 1.

Kermit says:
With primes, anything might be possible.

[Non-text portions of this message have been removed]
• ... This product is not very difficult to compute if you know about the function gamma (the extension of factorials to the complex domain). You are trying to
Message 1 of 11 , Dec 5 2:24 PM
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<mark.underwood@s...> wrote:
>
> --- In primenumbers@yahoogroups.com, "Jacques Tramu"
> <jacques.tramu@e...> wrote:
> >
> > I have computed the product and found
> > n= 200 000 000 : 0.9048530178
> > n= 1 000 000 0000 : 0.9048042986
> > n= 1 500 000 000 : 0.9047987573
> >
>
> I was just averaging some figures around the n = 40,000 area and it
> worked out to be around .9105. Looking a Jacque's findings the
> product seems to be ever so slowly shrinking.
>
> Contrast this the product 1/2 * 4/3 * 5/6 * 8/7 * ....
>
> and this appears to be converging to .599070...
>
> Mark
>

This product is not very difficult to compute if you know about the
function gamma (the extension of factorials to the complex domain).

You are trying to find:

k=inf (4k+1)(4k+4) k=inf 2
Prod ------------ = Prod 1 - ------------
k=0 (4k+2)(4k+3) k=0 (4k+2)(4k+3)

This is clearly convergent. But what is the limit?

k=n (4k+1)(4k+4) k=n (k+1/4)(k+1)
Prod ------------ = Prod -------------- =
k=0 (4k+2)(4k+3) k=0 (k+2/4)(k+3/4)

gamma(k+5/4) gamma(k+2) gamma(2/4) gamma(3/4)
= ------------------------- ---------------------
gamma(k+6/4) gamma(k+7/4) gamma(1/4) gamma(1)

= A(k) x B

where A(k) is the first fraction and B the second.

We are interested in the value of A(k) as k->inf.

Fortunately it turns out that the limit is 1. This can be seen by
using Stirling approximation.

Since gamma(1) = 1 and gamma(1/2) = sqrt(pi) we finally get:

k=inf (4k+1)(4k+4) sqrt(pi) * gamma(3/4)
Prod ------------ = ----------------------
k=0 (4k+2)(4k+3) gamma(1/4)