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On the distribution of the \(\mathrm{lcm}\) of \(k\)-tuples and related problems. (English) Zbl 1528.11100

Let \(\mathbb{N}\) be a set of positive integers, and \(\mathcal{P}\) be the set of prime numbers. Let \(\mathrm{lcm}(n_1,n_2,\ldots,n_k)\) denote the least common multiple of positive integers \(\{n_1,n_2,\ldots, n_k\}\), and let \(\{X_1^{(x)},X_2^{(x)},\ldots\}\), \(x\geqslant 2\), be a sequence of independent discrete uniform random variables with values on the set \(\{1,2,\ldots,x\}\).
The author of this paper studies the distribution of the least common multiple of positive integers in \(\mathbb{N}\cap[1,x]\). The obtained results are presented in three theorems. The first of those theorems asserts that for any fixed \(k\geqslant 2\) and any \(0<t\leqslant 1\) \[ \mathbb{P}\bigg(\,\frac{\mathrm{lcm}\big(X_1^{(x)},\ldots,X_k^{(x)}\big)}{x^k}>t\bigg)= \sum_{n\leqslant 1/t}\mathbb{P}\Big(R_k=\frac{1}{n}\Big)\int_{nt}^1\frac{(-\log z)^{k-1}}{(k-1)!}\,\mathrm{d}z+O_t\Big(\,\frac{\log^{k-1} x}{x}\Big), \] where \[ R_k=\prod_{p\in\mathcal{P}}p^{\max_{j\leqslant k} G_j(p)-\sum_{j\leqslant k}G_j(p)}\in\frac{1}{\mathbb{N}}, \] and \(\{G_1(p),G_2(p),\ldots, G_k(p)\}\) is a sequence of independent geometrically distributed random variables such that \[ \mathbb{P}\big(G_j(p)=m\big)= \Big(1-\frac{1}{p}\Big)\frac{1}{p^m}, m\in\{0,1,\ldots\}. \] The other two theorems of the paper deal with the behavior of the expectation \(\mathbb{E}\Big(\mathrm{lcm}\big(X_1^{(x)},\ldots,X_k^{(x)}\big)\Big)\).

MSC:

11N56 Rate of growth of arithmetic functions
11N60 Distribution functions associated with additive and positive multiplicative functions
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References:

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