Little exercise in analytic number theory

Question

I've found this exercise in a book, it's the following: "Show that for $x\geq 1$, for all $n\leq x$ except $o(x)$, the function $\Omega(n)$, which counts the number of prime divisors of $n$ with their multeplicity, is $(1+o(1))\log\log n$". How can I start? Thanks!

Answer 1

Hardy and Wright, An Introduction to the Theory of Numbers, 6th edition, Theorem 431 states that the number of $n$ not exceeding $x$ for which $$|\Omega(n)-\log\log n|>(\log\log n)^{.5+\delta}$$ is $o(x)$ for every positive $\delta$. I'm not going to write out the proof, as it's lengthy. The footnotes on page 498 are also quite informative.