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Re: [PrimeNumbers] Digest Number 3642

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  • Kermit Rose
    ... 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
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      On 2/14/2013 7:03 PM, primenumbers@yahoogroups.com wrote:
      > 1a. Re: Is the twin prime constant irrational?
      > 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


      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]
    • djbroadhurst
      ... http://arxiv.org/pdf/1004.4238.pdf Problem 3 ... Section 2.3 of same essay David
      Message 2 of 4 , Feb 15, 2013
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        --- 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
      • bobgillson
        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
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          Absolutely 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]
        • djbroadhurst
          ... 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
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            --- 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
            > 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
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