- View SourceOn 2/14/2013 7:03 PM, primenumbers@yahoogroups.com wrote:
> 1a. Re: Is the twin prime constant irrational?

Thank you David.

> Posted by: "djbroadhurst"d.broadhurst@... djbroadhurst

> 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

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] - View Source--- In primenumbers@yahoogroups.com,

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 - View SourceAbsolutely fascinating David, including your CV!

Thank you for sharing this paper with us.

Bob

On 15 Feb 2013, at 22:52, "djbroadhurst" <d.broadhurst@...> wrote:

> --- In primenumbers@yahoogroups.com,

> 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] - View Source--- In primenumbers@yahoogroups.com, bobgillson@... wrote:

> 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

David

> 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.