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Is this product convergent

  1. Apr 3, 2007 #1

    tpm

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    Following Euler if we define the product:

    [tex] (x-2^{-s})(x-3^{-s}) (x-5^{-s})(x-7^{-s}).......=f(x) [/tex]

    taken over all primes and s > 1 ,what would be the value of f(x) ?? i believe that [tex] f(x,s)=1/Li_{s} (x) [/tex] (inverse of Polylogarithm) however i'm not 100 % sure, although for x=1 you get the inverse of Riemann Zeta
     
    Last edited: Apr 3, 2007
  2. jcsd
  3. Apr 7, 2007 #2

    Gib Z

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    1. Inverse and Reciprocals aren't the same thing.

    2. For any value of s, and x > 1, the terms do not approach 1, but x. x being more than 1, the terms are not approaching 1, so the product does not converge.
     
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