I'm having a bit of a problem with exercise 4.12 in Apostol's "Introduction to Analytic Number Theory". I don't think it's supposed to be a very hard exercise, it's the first one in its section (they're usually a bit like warm-ups). I'm supposed to show that
If $a>0$ and $b>0$, then $\pi(ax)/\pi(bx) \sim a/b$ as $x \to \infty$.
It also says I'm allowed to use the prime number theorem. Is it just something like (a rough sketch): $\frac{\pi(ax)}{\pi(bx)} \sim \frac{ax \log bx}{bx \log ax} \sim \frac{a}{b}, \quad \text{since the logs $\to 1$ as $x \to \infty$?}$ I don't know, maybe I'm heading in the wrong direction... It would be very nice if someone could show me how to do this properly!