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I've heard that there is an easy way to derive the asymptotic $\prod_{p\le x} \left(1-\frac{1}{p}\right) \sim \frac{c}{\log(x)}$ if one isn't interested in deriving $c=e^{-\gamma}$. I don't see how to do this, however. Does anyone here know where I could find a simple proof of this statement or even write down a proof for me?

I'm quite new to number theory, so if you only assumed minimal background, that would be very helpful. Thanks for your help!

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    @RaymondManzoni: Thank you for the link!2012-10-26

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See pages 21-22 of Gérald Tenenbaum and Michel Mendès France, The Prime Numbers and Their Distribution. I found that by typing $\rm Mertens\ formula$ into the web. Many other possibly useful references came up, as well; for example, the discussion starting on page 88 of Hildebrand's notes.

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    Cheers! ${}{}{}{}$2012-10-25