## Re: [PrimeNumbers] Digest Number 3642

Expand Messages
• ... 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
Message 1 of 4 , Feb 15, 2013
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

[Non-text portions of this message have been removed]
• ... http://arxiv.org/pdf/1004.4238.pdf Problem 3 ... Section 2.3 of same essay David
Message 2 of 4 , Feb 15, 2013
Kermit Rose <kermit@...> wrote:

> how do we know what zeta(2) is

http://arxiv.org/pdf/1004.4238.pdf Problem 3

> and how do we know what zeta(4) is?

Section 2.3 of same essay

David
• Absolutely fascinating David, including your CV! Thank you for sharing this paper with us. Bob ... [Non-text portions of this message have been removed]
Message 3 of 4 , Feb 15, 2013
Absolutely fascinating David, including your CV!

Thank you for sharing this paper with us.

Bob

> Kermit Rose wrote:
>
> > how do we know what zeta(2) is
>
> http://arxiv.org/pdf/1004.4238.pdf Problem 3
>
> > and how do we know what zeta(4) is?
>
> Section 2.3 of same essay
>
> David
>
>

[Non-text portions of this message have been removed]
• ... Jack s ratio is the subject of a notable erratum: http://www.sciencedirect.com/science/article/pii/037026939500269Q ... David
Message 4 of 4 , Feb 15, 2013

> Absolutely fascinating David

Jack's ratio is the subject of a notable erratum:

http://www.sciencedirect.com/science/article/pii/037026939500269Q

which links to a .pdf file acknowledging a "pitiable" mistake:

> In our original computer program for the evaluation of the
> three-loop QCD correction to the rho parameter a pitiable mistake
> was overlooked. The wrong value 2/5 was substituted for
> zeta(2)^2/zeta(4) = 5/2.

David
Your message has been successfully submitted and would be delivered to recipients shortly.