Skip to Main content Skip to Navigation
Journal articles

On the least common multiple of several random integers

Abstract : Let $L_{n}(k)$ denote the least common multiple of $k$ independent random integers uniformly chosen in $\{1,2,\ldots ,n\}$. In this article, using a purely probabilistic approach, we derive a criterion for the convergence in distribution as $n\to \infty $ of $\frac{f(L_{n}(k))}{n ^{rk}}$ for a wide class of multiplicative arithmetic functions~$f$ with polynomial growth $r\in\mathbb{R}$. Furthermore, we identify the limit as an infinite product of independent random variables indexed by the set of prime numbers. Along the way, we compute the generating function of a trimmed sum of independent geometric laws, occurring in the above infinite product. This generating function is rational; we relate it to the generating function of a certain max-type Diophantine equation, of which we solve a generalized version. Our results extend theorems by Erd\H{o}s and Wintner (1939), Fern\'{a}ndez and Fern\'{a}ndez (2013) and Hilberdink and T\'{o}th (2016).
Complete list of metadata

Cited literature [28 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-01984389
Contributor : Alin Bostan <>
Submitted on : Thursday, January 17, 2019 - 12:57:37 AM
Last modification on : Friday, April 30, 2021 - 10:00:42 AM
Long-term archiving on: : Thursday, April 18, 2019 - 12:41:21 PM

File

1901.03002.pdf
Files produced by the author(s)

Identifiers

Citation

Alin Bostan, Alexander Marynych, Kilian Raschel. On the least common multiple of several random integers. Journal of Number Theory, Elsevier, 2019, 204, pp.113--133. ⟨10.1016/j.jnt.2019.03.017⟩. ⟨hal-01984389⟩

Share

Metrics

Record views

376

Files downloads

276