I am looking for approximations, or a closed form, if available, for the sum

$S(n,k)=\sum_{1\leq i,j,\leq n} lcm(i,j)^{-k}$

where lcm(i,j) is the least common multiple of integers $i,j$ and $k$ is a positive integer.

For the case $k=1$, I can see that a very loose lower bound is $3H_n-2$ where $H_n$ is the harmonic sum. Another lower bound is obtained by replacing $lcm(i,j)$ by $ij$ which yields $H_n^2$ for $k=1$ and $H_n^{2k}$ in general.

I am, however more interested in upper bounds, and small and even $k$ values

I am certainly no expert in number theory, so my apologies if this is trivial, but at least the statement looks clean to me.

Remark: There seems to be some work on determinants of matrices with entries of the form above.