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arXiv · 1801.08934

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

Gerold Alsmeyer·Zakhar Kabluchko·Alexander Marynych
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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 $θ\in (0,1)$ 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 $θ$ depends on $n$ in an appropriate way.

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Gerold Alsmeyer, Zakhar Kabluchko, Alexander Marynych. 2018-01-26. Limit theorems for the least common multiple of a random set of integers. https://arxiv.org/abs/1801.08934

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