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

Alsmeyer, Gerold; Kabluchko, Zakhar; Marynych, Alexander

doi:10.1090/tran/7871

Limit theorems for the least common multiple of a random set of integers
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by Gerold Alsmeyer, Zakhar Kabluchko and Alexander Marynych PDF

Trans. Amer. Math. Soc. 372 (2019), 4585-4603 Request permission

Abstract:

Let $L_{n}$ be the least common multiple of a random set of integers obtained from $\{1,\ldots ,n\}$ by retaining each element with probability $\theta \in (0,1)$ independently of the others. We prove that the process $(\log L_{\lfloor nt\rfloor })_{t\in [0,1]}$, 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 $\log L_{n}$ as well as Poisson limit theorems in regimes when $\theta$ depends on $n$ in an appropriate way.

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