According to Hardy-Ramanujan/Erdős-Kac we know that usually there are $\sim\log\log n$ prime numbers in a factorization. But if you pick up a natural number at random, and you factor it, what is the expected distribution of its factors?

By the Turán–Kubilius inquality, we see that this is the expectation: $$ \left\{\frac{\log \log p}{\log \log n}:\ p\mid n\right\}$$ tend to equidistribute on $[0,1]$.

Where can I find a more precise statement of the above?

[Edit: a previous version contained comments irrelevant to the question]


For precise information on many questions of this type you should consult the Cambridge Tract, Divisors by Hall and Tenenbaum. Here is Hildebrand's review in the Bulletin of this book. A much more precise version of your question is a result of Erdős. Suppose $n =p_1^{\alpha_1} \cdots p_k^{\alpha_k}$ is the prime factorization of $n$ with $p_1 < p_2 < \ldots <p_k$. Thus for almost all $n$, Hardy-Ramanujan tells us that $k=\omega(n) \sim \log \log n$. Let $\xi(n)$ be any function tending to infinity with $n$. For almost all $n$, Erdős showed that
$$ |\log \log p_i - i| \le (1+\epsilon) \sqrt{2i \log \log i} $$ holds for all $\xi(n)\le i \le k$; see (3) of Hildebrand's review. There has been more recent progress in the field, but Divisors is still a good place to start.

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    $\begingroup$ More recent progress? I'd be interested in knowing what has been added in excess of what lies in the first chapter of that book... $\endgroup$ – Captain Darling May 20 '17 at 21:56
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    $\begingroup$ I had in mind progress on things like the multiplication table problem, which is treated in the book. See the work of Ford, Koukoulopoulos and others. $\endgroup$ – Lucia May 21 '17 at 4:20

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