欧拉上/下数在奇素数幂处的模周期

中文摘要 AI 辅助

设$E_n$为$\{1,\dots,n\}$的交错排列数，等价于满足生成函数关系$\sum_{n\ge0}E_nz^n/n! = \sec z + \tan z$。对任意$q\ge1$，序列$(E_n\bmod q)_{n\ge0}$是最终周期的，记其最小最终周期为$d(q)$，预周期为$s(q)$。对任意奇素数$p$，Knuth与Buckholtz已证明$d(p)=\operatorname{lcm}(p-1,4)$，且满足$d(p^r)\mid p^{r-1}d(p)$、$s(p^r)\le r$；Ramassamy则猜想对所有$r\ge1$，这两个界均可达。本文中，我们在环$S_r=(\mathbb Z/p^r\mathbb Z)[x]/(x^2+1)$上引入欧拉数的代数频率展开，利用Hurwitz级数将欧拉序列代数地表示为有限个形式指数模的组合，其方式类似傅里叶分析。通过该展开，我们证明对任意奇素数$p$和任意$r\ge1$，$d(p^r)=p^{r-1}d(p)$，从而证实了Ramassamy的周期猜想；还通过证明$s(5^5)\le4<5$否定了预周期猜想；最后证明$5^5$是使$s(p^r)\ne r$的最小奇素数幂，并基于结果猜想对任意奇素数$p$和任意$r\ge2$，$s(p^r)\ge r-2$。

英文摘要

Let $E_n$ denote the number of alternating permutations of $\{1,\dots,n\}$, equivalently characterized by $\sum_{n\ge0}E_nz^n/n!=\sec z+\tan z$. For every $q\ge1$, the sequence $(E_n\bmod q)_{n\ge0}$ is eventually periodic; let $d(q)$ and $s(q)$ denote its minimal eventual period and preperiod. For every odd prime $p$, Knuth and Buckholtz proved $d(p)=\operatorname{lcm}(p-1,4)$ together with \[ d(p^r)\mid p^{r-1}d(p), \qquad s(p^r)\le r, \] and Ramassamy conjectured that both bounds are attained for every $r\ge1$. In this paper, we introduce an algebraic frequency expansion for the Euler zig-zag numbers using Hurwitz series over the coefficient ring $S_r=(\mathbb Z/p^r\mathbb Z)[x]/(x^2+1)$. More precisely, the corresponding Hurwitz series is represented as a finite combination of formal exponential modes, in a manner reminiscent of Fourier analysis. Using this expansion, we prove \[ d(p^r)=p^{r-1}d(p) \qquad \text{for every odd prime $p$ and every $r\ge1$}, \] thereby establishing Ramassamy's period conjecture. We also disprove the preperiod conjecture by proving \[ s(5^5)=4<5. \] Finally, we prove that $5^5$ is the smallest odd prime power for which $s(p^r)\ne r$, and based on our findings we further conjecture \[ s(p^r)\ge r-2 \] for every odd prime $p$ and every $r\ge2$.

发表机构

• Istanbul Technical University（伊斯坦布尔理工大学）

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