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24871Re: [PrimeNumbers] Digest Number 3642

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  • Kermit Rose
    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








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