论文标题
在随机$ q $ -Integers的最少常见倍数上
On the least common multiple of random $q$-integers
论文作者
论文摘要
For every positive integer $n$ and for every $α\in [0, 1]$, let $\mathcal{B}(n, α)$ denote the probabilistic model in which a random set $\mathcal{A} \subseteq \{1, \dots, n\}$ is constructed by picking independently each element of $\{1, \dots, n \} $带有概率$α$。 Cilleruelo，Rué，šarka和Zumalacárregui证明了$ \ Mathcal {a} $的元素中最不常见倍数的对数的几乎渐近公式。 令$ q $为不确定的，让$ [k] _q：= 1 + q + q^2 + \ cdots + q^{k-1} \ in \ mathbb {z} [q] $ $ be $ q $ q $ -Analog的正Integer $ k $。我们确定$ x：= \ operatorName {deg} \ protatorName {lcm} \！\ big（[\ Mathcal {a}] _ q \ big）$的期望值和方差\ Mathcal {a} \ big \} $。然后，我们证明了$ x $几乎可以肯定的渐近公式，这是Cilleruelo等人结果的$ Q $ -Analog。
For every positive integer $n$ and for every $α\in [0, 1]$, let $\mathcal{B}(n, α)$ denote the probabilistic model in which a random set $\mathcal{A} \subseteq \{1, \dots, n\}$ is constructed by picking independently each element of $\{1, \dots, n\}$ with probability $α$. Cilleruelo, Rué, Šarka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of $\mathcal{A}$. Let $q$ be an indeterminate and let $[k]_q := 1 + q + q^2 + \cdots + q^{k-1} \in \mathbb{Z}[q]$ be the $q$-analog of the positive integer $k$. We determine the expected value and the variance of $X := \operatorname{deg} \operatorname{lcm}\!\big([\mathcal{A}]_q\big)$, where $[\mathcal{A}]_q := \big\{[k]_q : k \in \mathcal{A}\big\}$. Then we prove an almost sure asymptotic formula for $X$, which is a $q$-analog of the result of Cilleruelo et al.
