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I have just been introduced the topic of distribution of primes, big O notation and aymptotic functions so please correct me if I say something that does not make sense. I am looking to get an intuitive meaning behind Mertens' estimates. The theorem is as follows

Suppose that $x \geq 2$. Then $$ \sum_{n \leq x} \frac{\Lambda(n)}{n} = \log x + O(1)$$ $$ \sum_{p \leq x} \frac{\log p}{p} = \log x + O(1)$$ for some constant $b$, $$\sum_{p \leq x} \frac 1 p = \log{\log{x}} + b + O \left( \frac{1}{\log{x}} \right) $$ for some constant $c>0$, $$\prod_{p \leq x} \left( 1 - \frac 1 p \right) = \frac{c}{\log x}\left( 1 + O\left(\frac{1}{\log x} \right) \right)$$

I am looking for intuitive meanings for all of them, more importantly the last one. By intuitive meaning, I mean that what do these equations represent? for example for the last one, I read the left hand side (the product) represents the density of positive integers that have no prime factor $\leq x$. But I don't understand how that means that so maybe in a bit easier terms.


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