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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).
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Submitted on : Thursday, January 17, 2019 - 12:57:37 AM
Last modification on : Friday, January 21, 2022 - 3:20:56 AM
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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⟩



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