Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

Thanks

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.