关于移位幂的最小公倍数
On the least common multiple of shifted powers
摘要：
令$a \geq 2$为整数。证明了对于$\{-1, +1\}$中的每个周期序列$(s_n)_{n \geq 1}$，存在一个有效的可计算有理数$C_\mathbf{s} > 0$使得\begin{equation*} \log\operatorname{lcm}(a + s_1, a^2 + s_2, \dots, a^n + s_n) \sim C_\mathbf{s} \cdot \frac{\log a}{π^2} \cdot n^2 , \end{equation*}为$n \to +\infty$，其中$\operatorname{lcm}$表示最小公倍数.此外，我们表明，如果$(s_n)_{n \geq 1}$是一个序列的独立和均匀分布的随机变量在$\{-1, +1\}$然后\begin{equation*} \log\operatorname{lcm}(a + s_1, a^2 + s_2, \dots, a^n + s_n) \sim 6 \operatorname{Li}_2\!\big(\tfrac1{2}\big) \cdot \frac{\log a}{π^2} \cdot n^2 , \end{equation*}概率$1 - o(1)$，作为$n \to +\infty$，其中$\operatorname{Li}_2$是双对数函数。
英文摘要：
Let $a \geq 2$ be an integer. We prove that for every periodic sequence $(s_n)_{n \geq 1}$ in $\{-1, +1\}$ there exists an effectively computable rational number $C_\mathbf{s} > 0$ such that \begin{equation*} \log\operatorname{lcm}(a + s_1, a^2 + s_2, \dots, a^n + s_n) \sim C_\mathbf{s} \cdot \frac{\log a}{π^2} \cdot n^2 , \end{equation*} as $n \to +\infty$, where $\operatorname{lcm}$ denotes the least common multiple. Furthermore, we show that if $(s_n)_{n \geq 1}$ is a sequence of independent and uniformly distributed random variables in $\{-1, +1\}$ then \begin{equation*} \log\operatorname{lcm}(a + s_1, a^2 + s_2, \dots, a^n + s_n) \sim 6 \operatorname{Li}_2\!\big(\tfrac1{2}\big) \cdot \frac{\log a}{π^2} \cdot n^2 , \end{equation*} with probability $1 - o(1)$, as $n \to +\infty$, where $\operatorname{Li}_2$ is the dilogarithm function.
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