I have a finite* set $A=a_1<\cdots<a_r$ of positive integers. Define $B$ as the set of positive integer multiples of $A$ and $$ A_1=\frac{1}{a_1} $$ $$ A_k=\frac{1}{a_k}-\sum_{i<j<k}\frac{1}{\operatorname{lcm}(a_i,a_j)}+\sum_{h<i<j<k}\frac{1}{\operatorname{lcm}(a_h,a_i,a_j)}-\cdots. $$
The density of $B$ is $A_1+\cdots+A_r.$ (If $A$ is infinite the density may not exist, but following the procedure above yields the logarithmic density, which (for sets of multiples) always equals the lower density.)
Is there any practical way to compute the sum above? Naive expansion of the inclusion-exclusion tree yields exponentially many terms.
* My set is actually infinite, but I'm lower-bounding the density of its multiples by the density of the multiples of a finite segment. If you can find the density of the set of multiples of an infinite set, this would also answer my question.
Edit: 14 hours left for the bounty!