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 note, 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 $r>-1$. Furthermore, we identify the limit as an infinite product of independent random variables indexed by 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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  • HAL Id : hal-01984389, version 1
  • ARXIV : 1901.03002

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Alin Bostan, Alexander Marynych, Kilian Raschel. On the least common multiple of several random integers. 2019. ⟨hal-01984389⟩

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