Let $\delta>0$. I am interested in obtaining a bound for the sum $\sum\_{1 \leq x\_1, ..., x\_n \leq N} \operatorname{lcm}(x\_1, ..., x\_n)^{- \delta}$ where lcm denotes the lowest common multiple of the numbers. I would appreciate any comments and suggestions! Thank you very much. 



How to bound $\sum_{1 \leq x_1, ..., x_n \leq N} lcm(x_1, ..., x_n)^{- \delta}$?
For $\delta>1$ the sum is bounded, while for $\delta=1$ it grows by a power of $\log N$.


So let me focus on $0<\delta<1$ and $N\geq 2$. For a simple lower bound, we have
$$ \sum\_{1 \leq x\_1, ..., x\_n \leq N} \operatorname{lcm}(x\_1, ..., x\_n)^{- \delta}\geq \left(\sum\_{1 \leq x \leq N} x^{- \delta}\right)^n\gg\_{n,\delta}N^{n(1-\delta)}. $$
For a simple upper bound, we have
$$ \sum\_{1 \leq x\_1, ..., x\_n \leq N} \operatorname{lcm}(x\_1, ..., x\_n)^{- \delta}\leq \sum\_{1\leq m\leq N^n}\frac{\tau(m)^n}{m^\delta}\ll\_{n,\delta}N^{n(1-\delta)}(\log N)^{2^n-1}.$$
So, up to a logarithmic factor, the sum grows like $N^{n(1-\delta)}$.


