Consider the ring $\mathbb{Z}_n$ for some positive integer $n > 1$. If $n$ is prime, then every nonzero element is a unit, and this question is trivial, so take $n$ to be composite.

What is known about the number of units from in $[k] := \{1, 2, 3, 4, \ldots k\}?$ Specifically, is it true that units are ``uniformly distributed'' in the sense that for some $\alpha \in (0,1)$ the number of units in $[\alpha n]$ is $\alpha \phi(n) (1 +o(1))$ where $\phi(\cdot)$ is the Euler-phi function. Is there anything known about the error (i.e., the $o(1)$) function- even assuming the Riemann Hypothesis or variants thereof?