# On the proof that $\phi(n)/n$ has a limit law

In this question, $$\mathbb{N}$$ denotes the set of positive integers. Also, $$\overline{\mathrm{d}}$$, and $$\mathrm{d}$$ means upper natural density, and natural densitiy respectively. (They are the propotion of a subset of $$\mathbb{N}$$ within $$\mathbb{N}$$, and $$\overline{\mathrm{d}}$$ is calculated through limsup.)

I am reading G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Third Edition. There, I found an exercise (Exe. 253 in Chap. III.2) to prove that $$\frac{\phi(n)}{n}$$ has a limit law.

Judging whether a given arithmetic function has a limit law is a difficult question, but there is a famous sufficient condition for a real-valued arithmetic function $$f$$ to have a limit law. Namely, there is a family of positive integers $$\{a_\epsilon(n)\}_{\epsilon >0, n\in\mathbb{N}}$$ such that:

1. $$\lim_{\epsilon\to\infty} \limsup_{T\to\infty} \overline{\mathrm{d}}\{n\in\mathbb{N}|a_\epsilon(n)>T\} = 0,$$
2. $$\lim_{\epsilon\to 0} \overline{\mathrm{d}}\{n\in\mathbb{N}|\hspace{1mm}|f(n)-f(a_\epsilon(n))|>\epsilon\} = 0,$$ and
3. For each $$a\in\mathbb{N},\mathrm{d}\{n\in\mathbb{N}|a_\epsilon(n) = a\}$$ exists.

In Exe. 253, $$f(n) = \frac{\phi(n)}{n},$$ and the author specifies $$a_\epsilon(n) = \prod_{p^m||n, p\leq\epsilon^{-2}} p^m.$$The most difficult part of Exe. 253 seems to be (d), where you are invited to show the condition 2. above. I got stuck here, and am looking for solution. Any help is appreciated.