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.

 

[tacman]

function

Based on the information provided, it is difficult to determine if this product is convergent. Euler's definition of the product does not explicitly state if it is convergent or not. Additionally, the value of f(x) cannot be determined without knowing the specific values of x and s.

However, based on the given information, it seems that the product may have a value of 1/Li_s(x), which is the inverse of the Polylogarithm function. This function is closely related to the Riemann Zeta function, which has been extensively studied in mathematics and has a value of 1 when x=1. Therefore, if x=1, then the product may have a value of 1, which is the inverse of the Riemann Zeta function.

In order to determine if this product is convergent, more information is needed such as the specific values of x and s, and if the product has a finite limit as s approaches infinity. Additionally, further analysis and mathematical techniques may be necessary to determine the convergence of this product.