Limit theorems for the least common multiple of a random set of integers

الأصل · EN

Let Lₙ be the least common multiple of a random set of integers obtained from {1,,n} by retaining each element with probability θ∈ (0,1) independently of the others. We prove that the process (L nt)ₜ∈ [₀,₁], after centering and normalization, converges weakly to a certain Gaussian process that is not Brownian motion. Further results include a strong law of large numbers for Lₙ as well as Poisson limit theorems in regimes when θ depends on n in an appropriate way.