Property of Riemann zeta with function of representation of a number n

Question

In the study of the properties of Rieamann zeta I encounterd the following:

Being $f_{k}(n)$ a multiplicative function as the number of representation of n as a product of k factor each greater than the unity when n>1 the order of the factor being essential. Then

$A:= \sum_{n=2}^{\infty} \frac{f_{k}(n)}{n^s} = {(\zeta(s)-1)}^{k}$ for $\sigma >1$

the book also remind the theorem

$\sum_{n=1}^{\infty} \frac{f(n)}{n^s} = \prod (1 + \frac{f(p)}{p^s} + \frac{f(p^2)}{p^{2s}} + \cdots)$

Does some one can deduce A?

Answer 1

The Wikipedia article Dirichlet series states:

Dirichlet series can be used as generating series for counting weighted sets of objects with respect to a weight which is combined multiplicatively when taking Cartesian products.

The primary example is the Riemann zeta function $\,\zeta(s)\,$ where each positive integer has the same weight $1$. If we exclude the integer $1$, the generating series is now $\,\zeta(s)-1.\,$ If we now multiply this generating series by itself $\,k\,$ times, then we are counting the number of representation of $\,n\,$ as a product of $\,k\,$ factors each greater than the unity. Each series factor of $\,\zeta(s)-1\,$ contributes one integer to the ordered integer factorization.

Answer 2

I think is easy to see if

$\zeta(s) -1 = \frac{1}{2^s} + \frac{1}{3^s} +\frac{1}{4^s} + \cdots$

therefore

$(\zeta(s) -1)^k = (\frac{1}{2^s} + \frac{1}{3^s} + \cdots)_1 \cdots (\frac{1}{2^s} + \frac{1}{3^s} + \cdots)_k $

then each f_{n} is the number of rapresentation of products of a number n.