# Confusing proof of brun's theorem?

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 you guys 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 ? Im 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 independant proof, so NO " if Riemann Hypothesis is correct then " or such.

An understandable independant proof, thats what I want.

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Hmmm... This is the second part only. The first is a little earlier page 100 and starts with the sieve ('crible' in french). –  Raymond Manzoni Jun 17 '13 at 23:00
Thanks. that might help. –  mick Jun 17 '13 at 23:51
@RaymondManzoni still confused. –  mick Jun 18 '13 at 0:00