The author of *Riemann's Zeta Function*, H.M.Edwards, says:

According to Euler, $\sum_{p<x}\frac{1}{p}\sim \log(\log(x))$ when $x\longrightarrow\infty$.

$\log(\log(x))=\int_{1}^{\log(x)} \frac{du}{u}=\int_{e}^{x} \frac{dv}{v\log(v)}$

so (1) says that the integral of $\frac{1}{v}$ relative to the measure $\frac{dv}{\log(v)}$ diverges in the same way as the integral of $\frac{1}{v}$ relative to the point mesaure which assigns weight $1$ to primes and weight $0$ to all other points. In this sense, **(1) can be regarded as saying that the density of primes is roughly $\frac{1}{\log(v)}$.**

And this is what the author says. I know that density formula for a number $x$ gives us the probability for a number $y<x$ of being prime, but I don't how the author identified $\frac{1}{\log(v)}$ as the density formula (it's an approximation of the actual density formula, I know it). He implies that it is a trivial reasoning, but I can't see it.

I think it could be because of the language: I'm a Spanish student, so my English is not too good and this is the first time that I read something like "the integral of $\frac{1}{v}$ relative to the measure $\frac{dv}{\log(v)}$ diverges in the same way…". I am not clear what it means, so I want to use this question for two things: to find out how the author identified the density formula, and to learn some technical English vocabulary.

Thanks.