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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
    
Maybe stuff about this "merlin sieve" would help. –  mick Jun 18 '13 at 0:02
    
I own a book by Tenenbaum who should make these things clearer (but I didn't read it yet so that I'll be of little help here... :-)). The book is Tenenbaum's 'Introduction to Analytic and Probabilistic Number Theory' (the french edition is much cheaper...) and most of the number theory stuff required is introduced at the start (Möbius inversion and so on..). You may try to start too with Wikipedia, Sebah and Gourdon's paper at the end (without the details of the not so easy proof I fear...). –  Raymond Manzoni Jun 18 '13 at 21:24

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