Take the tour ×  Let $\omega(n)$ be the number of distinct
primes dividing $n$.    1. For $x\in(0,1)$, let $\varphi(x,n)$
be the number of positive integers $m\leq xn$ which are prime to
$n$. Show that $\varphi(x,n)=x\varphi(n)+O(\tau(n))$.   2.
Deduce that as $n\to\infty,\,\varphi(x,n) \sim x\varphi(n)$.
This is an exercise in the book Fundamentals of Number Theory
(page 142). I was wondering how to connect it with $\omega(n)$.
Could you explain it to me or give a proof on that? Thanks in
advance.  share|improve this question There is a typo in your
post. –  Kerry Dec 9 '11 at 5:22 As written, this can't possibly
be right. Surely the $\phi(x)$ is meant to be $\phi(n)$. But
also $\phi(x,n)\to x\phi(n)$ doesn't make sense. Maybe
$\phi(x,n)$ is asymptotic to $x\phi(n)$? –  Gerry Myerson Dec 9
'11 at 8:16 @GerryMyerson Can you explain more about the second
problem? –  Kou Dec 19 '11 at 22:17 I think I'll echo Greg's
answer. It's really not hard to get from the 1st problem to the
2nd, if you know what that symbo