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.


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