I read Brun's proof of Brun's theorem here :
http://gallica.bnf.fr/ark:/12148/bpt6k486270d/f110.image (and the following pages)
and here
http://gallica.bnf.fr/ark:/12148/bpt6k486270d/f138.image (and the following pages)
But I was unsatisfied. I did not understand it and I did not even get the notation he used. It seems like " some statistical arguments " because of the infinite products used.
Could someone please explain the proof to me step by step? I understand that Brun's constant converges if the prime twins are $O(\dfrac{x}{\ln(x)^2})$ or $O(\dfrac{x\cdot \ln(\ln(x))}{\ln(x)^2})$, but apart from that he lost me from the beginning. Also I did not see a sieve or should I have seen it ? I'm new to sieve theory.
Edit
I would like to add that the conditions for the sieve are also very important to me; they need to be proven. A proof without a sieve would also be nice if possible.
I want an independent proof, so NO "if Riemann Hypothesis is correct then" or such.