I'm reading through The normal number of prime factors of a number $n$. I'm confused by a remark on the second page: let $f(n)$ represent the number of distinct prime factors of $n$. Then
we can shew (by purely elementary reasoning) that, if $\epsilon$ is any positive number, we have
$$ f(n) < (1 + \epsilon) \frac{\log n}{\log \log n}$$ for all sufficiently large values of $n$ and $$ f(n) > (1 - \epsilon) \frac{\log n}{\log \log n}$$ for an infinity of values; so that the maximum order of $f(n)$ is $$ \frac{\log n}{\log \log n} $$
I don't follow their "purely elementary reasoning." What am I missing?
You can use the facts that $\forall\epsilon>0$ holds $$1<\frac{\pi\left(x\right)\log\left(x\right)}{\theta\left(x\right)}<1+\epsilon$$ for all sufficiently large values of $x$ . Now if we take a sufficiently large $n$ of the form $$n=p_{1}p_{2}\cdots p_{k}$$ we have$$f\left(n\right)=k=\pi\left(p_{k}\right)$$ and$$\theta\left(p_{k}\right)=\underset{i\leq k}{\sum}\log\left(p_{i}\right)=\log\left(n\right).$$