Probabilistic Proofs of Key Number-Theoretic Results

Question

Given a positive integer $n$, let $p$ be the largest prime less than or equal to $n$.

Let $N(n)=2^{C_2}\cdots p^{C_p}$ be uniformly distributed from $1$ to $n$, and $M(n)=2^{Z_2}\cdots p^{Z_p}$ where $Z_p's$ are independent geometric with $P(Z_p\ge k)=\frac{1}{p^k}$. It can be shown that $C_p$ has probability mass $P(C_p=k)=\frac{\big\lfloor \frac{n}{p^k}\big\rfloor}{n}.$

Consider the metric $d(M,N)=\sum_{p\le n}\vert C_p-Z_p\vert$
($d$ counts the number of prime insertions and deletions needed to convert $N$ to $M$).

Using this metric, we have the Wasserstein metric $d_W(M,N)=\inf_{\text{couplings}}\mathbb{E}d(M,N).$

Arratia (page 10 of https://arxiv.org/pdf/1305.0941.pdf) claims

• $d_W(M,N)=o(\log \log n)$ implies the Hardy-Ramanujan Theorem (which states that for almost every positive integer $n$, $\omega(n) \approx \log \log n$, where $\omega(n)$ is the number of distinct prime divisors of $n$).
• $d_W(M,N)=o(\sqrt{\log \log n})$ implies the Erdos-Kac Central Limit Theorem (which states that $\frac{\omega(n)-\log \log n}{\sqrt{\log \log n}}$ has the standard normal distribution.

Further confirmation of this is stated by Bollobas on page 29 of "Contemporary Combinatorics":

How can these results be derived from the asymptotics?

Answer 1

Obviously, one has $$|\omega(M)-\omega(N)|\leq d(M,N).$$ Taking the expectations, we see that for every $n$, we can construct a coupling of $M(n)$ and $N(n)$ such that $$ \mathbb{E}|\omega(M(n))-\omega(N(n))|\leq d_W(M(n),N(n))+\frac{1}{n}. $$ Under the asymptotic estimates above, Chebyshev's inequality implies, for any $\alpha>0$ $$ \mathbb{P}(|\omega(M(n)-\omega(N(n)))|>\alpha \log \log n)=o(1), $$ or, respectively, $$ \mathbb{P}(|\omega(M(n)-\omega(N(n)))|>\alpha \sqrt{\log \log n})=o(1). $$ Hence, Hardy-Ramanujan and Erdos-Kac boil down to the corresponding statements about $\omega(M(n))$. These are just the LLN and the CLT for the independent random variables $$\mathbf{1}_{Z_2\neq 0},\mathbf{1}_{Z_3\neq 0},\mathbf{1}_{Z_5\neq 0}\dots,$$ which are of course well known and standard (e. g., Lindeberg's CLT applies)