Erdős [Er64d] proved that if $2^{r-1}<k\leq 2^r$ then \[g_k(n) \sim \frac{(\log\log n)^{r-1}}{(r-1)!\log n}n\] (which is the asymptotic count of those integers $\leq n$ with $r$ distinct prime factors).

In particular the asymptotics of $g_k(n)$ are known; in this problem Erdős was asking about the second order terms. For $k=3$ he could prove the existence of some $0<c_1\leq c_2$ such that \[\frac{\log\log n}{\log n}n+c_1\frac{n}{(\log n)^2}\leq g_k(n)\leq \frac{\log\log n}{\log n}n+c_2\frac{n}{(\log n)^2}.\]

The special case $k=2$ is the subject of [425].