Factoring of the partition function In Atkins physical chemistry pg 615, I found the following:

I did not understand how it is valid, since sigma(xy) is not equal to sigma(x) × sigma(y).
 A: Let me illustrate this with a simpler example. I think if you accept the following than you shouldn't have any trouble believing the formula in the book. ( I assume here that all the sums are finite, so that we don't care about convergence issues).
$$ \left(\sum_{i=1}^n e^{a_i} \right) \cdot \left( \sum_{j=1}^m e^{b_j} \right)= \sum_{i=1}^n \sum_{j=1}^m e^{a_i + b_j}  $$
where of course $a_i,b_j \in \mathbb R$. Note that you can write $e^{a_i + b_j}$ as $e^{a_i} \cdot e^{b_j}$. Using this and starting with the right-hand side we have:
$$ \sum_{i=1}^n \sum_{j=1}^m e^{a_i + b_j} = \sum_{i=1}^n \sum_{j=1}^m e^{a_i} \cdot e^{b_j} \overset {(*)} =  \left(\sum_{i=1}^n e^{a_i} \right) \cdot \left( \sum_{j=1}^m e^{b_j} \right) $$
where I used in $(*)$ the fact that the second summation is over $j$ so that $a_i$ are just constants for this summation. It is like writing $ 2 b_1 + 2 b_2 = 2 \cdot (b_1 + b_2)$ Thus you get the equality that you want.
A: Let Y equal X+1. So does sigma(x(x+1)) equal sigma(x)*sigma(x+1)? 
Clearly, NO.
So the best method is to keep it as a product. 
Summation doesn't work like that.
