On the least common multiple of several random integers

Alin Bostan (1) , Alexander Marynych (2) , Kilian Raschel (3)

1 SPECFUN - Symbolic Special Functions : Fast and Certified
2 Faculty of Cybernetics [Kyiv]
3 IDP - Institut Denis Poisson

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).

Mots clés: least common multiple; trimmed sums of geometric laws; prime products; Convergence in distribution

Domaines: Probabilités [math.PR]; Théorie des nombres [math.NT]

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