## 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

### Keywords:

distribution; least common multiple; tuples
Full Text:

### References:

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