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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: Gerold Alsmeyer, Zakhar Kabluchko and Alexander Marynych
Journal: Trans. Amer. Math. Soc. 372 (2019), 4585-4603
MSC (2010): Primary 60F05; Secondary 11N37, 60F15
DOI: https://doi.org/10.1090/tran/7871
Published electronically: July 2, 2019
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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 $ \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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Additional Information

Gerold Alsmeyer
Affiliation: Institute of Mathematical Stochastics, Department of Mathematics and Computer Science, University of Münster, Orléans-Ring 10, D-48149, Münster, Germany
Email: gerolda@uni-muenster.de

Zakhar Kabluchko
Affiliation: Institute of Mathematical Stochastics, Department of Mathematics and Computer Science, University of Münster, Orléans-Ring 10, D-48149, Münster, Germany
Email: zakhar.kabluchko@uni-muenster.de

Alexander Marynych
Affiliation: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine
Email: marynych@unicyb.kiev.ua

DOI: https://doi.org/10.1090/tran/7871
Keywords: Random set of integers, least common multiple, law of large numbers, central limit theorem, functional limit theorem, Gaussian process, von Mangoldt function, Chebyshev functions
Received by editor(s): January 26, 2018
Published electronically: July 2, 2019
Additional Notes: The first and second authors were partially supported by the Deutsche Forschungsgemeinschaft (SFB 878)
The third author was partially supported by the Alexander von Humboldt Foundation
Article copyright: © Copyright 2019 American Mathematical Society