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

2018/01/26 by Alsmeyer, Gerold, Kabluchko, Zakhar, Marynych, Alexander
#60F05 (Primary) 11N37 #60F15 (secondary) #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.1801.08934

Abstract

Let L_n 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 (log L_{\lfloor nt\rfloor})_{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 log L_n as well as Poisson limit theorems in regimes when θ depends on n in an appropriate way.

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