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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.

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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?

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  • $\begingroup$ It does, that makes a lot more sense. I feel like my intuition for number theory isn't built up very well. I've gone through elementary number theory, but I suppose I could use some better intuition on the stuff, any recommendations for material that might help with that? $\endgroup$
    – Vedvart1
    Apr 13, 2017 at 17:20
  • $\begingroup$ @Vedvart1 I am just a sophomore in college taking my first number theory course, but that subject has always come naturally to me, plus I have read a lot of number theory over the years for competitions. I generally have found that most of number theory can be understood either by factoring things or using congruences. A good overview of Euler's work in number theory would provide a great background, or any expository work you could find in a place like Barnes and Nobles would also be a good resource. $\endgroup$
    – Will Craig
    Apr 13, 2017 at 17:25
  • $\begingroup$ I think I'll pour over Euler's work then, and try and look at things as no more than a bunch of numbers and basic algebra whenever possible. I'm still only a junior in high school and I've taught myself most of what I know, so it's entirely possible I might just need to let the basics simmer in the back of my mind for awhile. Thank you for the help though, I always love when this site and its community pull through! $\endgroup$
    – Vedvart1
    Apr 13, 2017 at 17:28
  • $\begingroup$ Yeah I'd focus on linear congruences and the Chinese Remainder theorem before you get anywhere else. Those should provide a good background if you aren't yet familiar enough with congruences. $\endgroup$
    – Will Craig
    Apr 13, 2017 at 17:30

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