Does Euler's Product Over Primes Converge for s > 1?

[tpm]

Following Euler if we define the product:

$$(x-2^{-s})(x-3^{-s}) (x-5^{-s})(x-7^{-s})...=f(x)$$

taken over all primes and s > 1 ,what would be the value of f(x) ?? i believe that $$f(x,s)=1/Li_{s} (x)$$ (inverse of Polylogarithm) however I'm not 100 % sure, although for x=1 you get the inverse of Riemann Zeta

 

[Gib Z]

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.