1
$\begingroup$

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!

$\endgroup$
2
  • $\begingroup$ Perhaps that this thread will help. $\endgroup$ – Raymond Manzoni Oct 25 '12 at 22:08
  • $\begingroup$ @RaymondManzoni: Thank you for the link! $\endgroup$ – Sam Oct 26 '12 at 10:36
4
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Cheers! ${}{}{}{}$ $\endgroup$ – Sam Oct 25 '12 at 22:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.