# Looking for intuition behind the Euler Product Formula and related Number Theory things

I've been trying to dive into Number Theory to better understand prime numbers. I found Euler's Product Formula, $$\sum_{n\in\mathbb{N}}{\frac{1}{n^s}}=\prod_{\text{Prime }p}{\frac{1}{1-\frac{1}{p^s}}}$$ While looking into this, a number of things confused me. In one informal explanation, I don't get how they Add for n=1,2,3... on the very first page. In another book I have, I see them "expand the factors" by saying $$\frac{1}{1-\frac{1}{p^s}}=1+\frac{1}{p^s}+\frac{1}{(p^2)^s}+\frac{1}{(p^3)^s}+...$$ and I don't see where that comes from either. I'm also curious why we express the product as such an unwieldy fraction instead of simply writing $\frac{p^s}{p^s-1}$. I suppose my overarching question is... am I just blind and this isn't clicking with me, or am I in over my head and these steps involve other parts of number theory which aren't mentioned? I apologize if I've been too broad or not very specific, I'd be happy to clarify what I can or move this somewhere more appropriate.

Remember that $\dfrac{1}{1-x} = 1 + x + x^2 + ...$. If we let $x = 1/p$, we get a sequence of all inverse powers of $p$. Now, if we multiply these sequences over all $p$, we are generating every possible combination of various primes to various powers (distributive law) and by the Fundamental Theorem of Arithemtic we can know for sure that every inverse integer is expressed once and exactly once. Does that help?