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

MathSciNet review:
4009436

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the least common multiple of a random set of integers obtained from by retaining each element with probability independently of the others. We prove that the process , 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 as well as Poisson limit theorems in regimes when depends on 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