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Show by a sieve argument that the number of square free integers not exceeding $x$ is less than $$x\prod_p\left(1-\frac{1}{p^2}\right)+o(x),$$where the product extends over all primes.

I happened to see this exercise this morning, and still fail to prove it. Could you give me a proof?

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Do you mean "sum" in "where the product extends"? – Dimitrije Kostic Dec 8 '11 at 2:42
@DimitrijeKostic Sorry, it should be product. – Kou Dec 8 '11 at 2:51
@AndréNicolas Sorry, you are right. – Kou Dec 8 '11 at 2:52

Rough sketch: Have you seen the Wikipedia section on the distribution of squarefree integers? It essentially gives your solution. All you need to notice is that for large $n$, the "probability" that an integer has $p^2$ as a factor is $\frac{p^2-1}{p^2} = \left( 1 - \frac{1}{p^2}\right)$. Since the probability that $n$ has $p^2$ as a factor is roughly independent of the fact it has $q^2$ where $p$ and $q$ are distinct primes, this implies that

$$ \frac{Q(x)}{x} \approx \prod_p \left( 1 - \frac{1}{p^2} \right). $$

I will leave it to you to make this precise.

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