I was seeing the question Here
So, I have a doubt at the last line of the particular question. Let me frame in the same way wolf has done it, the main objective is to prove the same thing of Marius Overholt:
$Var[w]=O(\log\log(N))$ where $w(n)$ is the no of prime divisors of $n$.
There I am stuck at the point from
$N(\log\log(N))^2-N\sum_{p \leq \sqrt N} \frac 1{p}\sum_{n/p <q \leq N} \frac 1{q}-N\sum_{ \sqrt N < p \leq N} \frac 1{p}\sum_{n/p <q \leq N} \frac 1{q}+O(N\log\log(N))=^?N(\log\log(N))^2+O(N\log\log(N))$
Let me tell the reader one thing that here we have proved one result beforehand that $\sum_{p \leq N}\frac 1p=\log\log(N)+a+O\big(\frac 1{\log(N)}\big)$, where $a$ is a fixed constant.
This is a consequence of the Erdos-Kac theorem.
Quoting from wikipedia:
"states that if ω(n) is the number of distinct prime factors of n, then, loosely speaking, the probability distribution of ${\displaystyle {\frac {\omega (n)-\log \log n}{\sqrt {\log \log n}}}} $ is the standard normal distribution. This is an extension of the Hardy–Ramanujan theorem, which states that the normal order of ω(n) is log log n with a typical error of size ${\displaystyle {\sqrt {\log \log n}}} . $
Proofs are in the references here:
https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem