On the least common multiple of random q-integers

For every positive integer n and for every α∈[0,1], let B(n,α) denote the probabilistic model in which a random set A⊆{1,…,n} is constructed by picking independently each element of {1,…,n} with probability α. Cilleruelo, Rué, Šarka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of A.Let q be an indeterminate and let [k]q:=1+q+q2+⋯+qk−1∈Z[q] be the q-analog of the positive integer k. We determine the expected value and the variance of X:=deglcm([A]q), where [A]q:={[k]q:k∈A}. Then we prove an almost sure asymptotic formula for X, which is a q-analog of the result of Cilleruelo et al.