Is this product convergent

  • Thread starter tpm
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  • #1
tpm
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Main Question or Discussion Point

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
 
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Answers and Replies

  • #2
Gib Z
Homework Helper
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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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