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

$S(n,a,b)=\sum_{1\leq x,y,\leq n} \frac{x^a}{\mathrm{lcm}(x,y)^{b}}$

where $\mathrm{lcm}(x,y)$ is the least common multiple of integers $x,y$ and $a,b$ are positive quantities. I am in particular interested in $a=b=1.$ For this case numerical evidence suggests $$ S(n,1,1)=O( n \log n) $$ may hold. In particular, I am wondering whether by using the technique in the answer to this question here, one might obtain (as $n \rightarrow \infty$), by letting $a,b\downarrow 1,$ an estimate in terms of zeta functions. In that question the upper bound $$ S(n,0,b)\leq\frac{\zeta(b)^3}{\zeta(2b)},\quad b>1 $$ is derived by letting $n\rightarrow \infty.$ Any pointers,comments welcome.

I am looking for approximations, or a closed form, if available, for the sum
$$S(n,a,b)=\sum_{1\leq x,y,\leq n} \frac{x^a}{\mathrm{lcm}(x,y)^{b}},$$ where $\mathrm{lcm}(x,y)$ is the least common multiple of integers $x,y$ and $a,b$ are positive quantities. I am in particular interested in $a=b=1.$ For this case numerical evidence suggests $$ S(n,1,1)=O( n \log n) $$ may hold. In particular, I am wondering whether by using the technique in the answer to this question here, one might obtain (as $n \rightarrow \infty$), by letting $a,b\downarrow 1,$ an estimate in terms of zeta functions. In that question the upper bound $$ S(n,0,b)\leq\frac{\zeta(b)^3}{\zeta(2b)},\quad b>1 $$ is derived by letting $n\rightarrow \infty.$ Any pointers,comments welcome.