Define $\phi(n,x)= \sum_{m\leq x,\gcd(m,n)=1} 1$, the number of elements in the interval $[1,x]$ that is relatively prime to $n$. $\omega(n)$ is the number of distinct prime factors of $n$.

It's not difficult to show that $\phi(n,x) \geq x \phi(n)/n - 2^{\omega(n)}$ (see this question). This shows for large $x$, say $x\geq n^\frac{1}{\log \log n}$, $\phi(n,x)$ is essentially $x\phi(n)/n$.

Are there better bounds known for smaller values of $x$, say $x=o(2^{\omega(n)})$? The bound is negative for $x$ in those values.

Even the naive bound $\phi(n,x)\geq 1+\pi(x)-\omega(n)$ performs better when $x$ is small.