arXivDaily arXiv每日学术速递 周一至周五更新
arXiv周末暂无论文更新，休息一下吧，周末愉快~~
arXiv 1909.04342math.NT
收藏

在液晶显示器上。二元递归序列随机项的

On the l.c.m. of random terms of binary recurrence sequences

Carlo Sanna

发表信息：
Journal of Number Theory 213 (2020), 221-231
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

对于每个正整数$n$和每个$δ\in [0,1]$，令$B(n, δ)$表示概率模型，其中通过以概率$δ$独立地选择$\{1, \dots, n\}$的每个元素来构造随机集$A \subseteq \{1, \dots, n\}$。此外，令$(u_k)_{k \geq 0}$是满足$u_k = a_1 u_{k - 1} + a_2 u_{k - 2}$的整数序列，对于每个整数$k \geq 2$，其中$u_0 = 0$，$u_1 \neq 0$和$a_1, a_2$是固定的非零整数;并且令$α$和$β$，以及$|α| \geq |β|$，是多项式$X^2 - a_1 X - a_2$的两个根。此外，假设$α/ β$不是单位根。我们证明，作为$δn / \log n \to +\infty$，对于$B(n, δ)$中的每个$A$，我们都有概率为$1 - o(1)$的$$\log \operatorname{lcm} (u_a : a \in A) \sim \frac{δ\operatorname{Li}_2(1 - δ)}{1 - δ} \cdot \frac{3\log\!\big|α/ \!\sqrt{(a_1^2, a_2)}\big|}{π^2} \cdot n^2 $$，其中$\operatorname{lcm}$表示最低公倍数，$\operatorname{Li}_2$是双对数，并且涉及$δ$的因子意味着当$δ= 1$时等于$1$。这扩展了Akiyama，Tropak，Matiyasevich，Guy，Kiss和Mátyás以前的结果，他们研究了确定性情况$δ= 1$，并且受到Cilleruelo，Rué，Šarka和Zumalacárregui的$\operatorname{lcm}(A)$的渐近公式的启发。

英文摘要：

For every positive integer $n$ and every $δ\in [0,1]$, let $B(n, δ)$ denote the probabilistic model in which a random set $A \subseteq \{1, \dots, n\}$ is constructed by choosing independently every element of $\{1, \dots, n\}$ with probability $δ$. Moreover, let $(u_k)_{k \geq 0}$ be an integer sequence satisfying $u_k = a_1 u_{k - 1} + a_2 u_{k - 2}$, for every integer $k \geq 2$, where $u_0 = 0$, $u_1 \neq 0$, and $a_1, a_2$ are fixed nonzero integers; and let $α$ and $β$, with $|α| \geq |β|$, be the two roots of the polynomial $X^2 - a_1 X - a_2$. Also, assume that $α/ β$ is not a root of unity. We prove that, as $δn / \log n \to +\infty$, for every $A$ in $B(n, δ)$ we have $$\log \operatorname{lcm} (u_a : a \in A) \sim \frac{δ\operatorname{Li}_2(1 - δ)}{1 - δ} \cdot \frac{3\log\!\big|α/ \!\sqrt{(a_1^2, a_2)}\big|}{π^2} \cdot n^2 $$ with probability $1 - o(1)$, where $\operatorname{lcm}$ denotes the lowest common multiple, $\operatorname{Li}_2$ is the dilogarithm, and the factor involving $δ$ is meant to be equal to $1$ when $δ= 1$. This extends previous results of Akiyama, Tropak, Matiyasevich, Guy, Kiss and Mátyás, who studied the deterministic case $δ= 1$, and is motivated by an asymptotic formula for $\operatorname{lcm}(A)$ due to Cilleruelo, Rué, Šarka, and Zumalacárregui.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑