## 24871Re: [PrimeNumbers] Digest Number 3642

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• Feb 15, 2013
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On 2/14/2013 7:03 PM, primenumbers@yahoogroups.com wrote:
> 1a. Re: Is the twin prime constant irrational?
> Date: Wed Feb 13, 2013 5:14 pm ((PST))
>
>
>
> --- Inprimenumbers@yahoogroups.com, Jack Brennen wrote:
>> >
>> >What is the product over all of the primes p of:
>> >
>> > (p^2+1)/(p^2-1) ?
>> >
>> >That's a constant that requires EVERY prime in order
>> >to calculate it.
>> >
>> >It turns out to be 5/2. Which is not irrational.
> Nice point, Jack.
>
> print(zeta(2)^2/zeta(4));
> 2.5000000000000000000000000000000000000
>
> David

Thank you David.

I see how (zeta(2))^2/zeta(4) = (p^2+1)/(p^2-1),

but how do we know what zeta(2) is, and how do we know what zeta(4) is?

zeta(2) = sum(k positive integer)(1/k^2)
= product(p positive prime, J non-negative integer)(sum(1/p^(2J))

= product(p positive prime)(1/(1-1/p^2))

= product(p positive prime)(p^2/(p^2-1))

zeta(4) = sum(k positive integer)(1/k^4)
= product(p positive prime, J non-negative integer)(sum(1/p^(4J))

= product(p positive prime)(1/(1-1/p^4))

= product(p positive prime)(p^4/(p^4-1))

(zeta(2))^2/zeta(4)
= product(p positive prime)(((p^2/(p^2-1)))^2/(p^4/(p^4-1)))

= product(p positive prime)((p^4/(p^2-1)^2)/(p^4/(p^4-1)))

= product(p positive prime)((p^4-1)/(p^2-1)^2))

= product(p positive prime)((p^2+1)/(p^2-1))

Kermit

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