Let $\Lambda(x,n)$ be the the number of totatives of $x$ which are less than or equal to $n$, and $\Phi(x)$ be Euler's totient function.

For now assume $x>n$.


>Is there a general formula for  $\Lambda(x,n)$ ?




Let $l = gcd(x,n)$, $x'= x/l$ and $n'=n/l$

I have a proof in my head that...


$\Lambda(x,n) = \frac{n'}{x'} \Phi(x) \pm V$, where the variance $ 0 \leq V \leq \frac{x'-n'}{x'n'} \Phi(x)$