互反幂和与广义莱默型乘积的高阶同余
Higher-Order Congruence for Reciprocal Power Sums and Generalized Lehmer-Type Products
AI总结：
研究互反幂和与广义莱默型乘积的高阶同余，将互反平方和结果扩展到奇数次互反和并建立模\(n\)统一同余，通过完全指数贝尔多项式解决广义莱默型乘积高阶同余无简单闭式问题，提供相关显式计算与验证框架。
AI中文摘要：
本文研究互反幂和与莱默型乘积的高阶同余。设\(n\geq1\)且\((n,6)=1\)，\(e\in\{2,3,4,6\}\)。对于互反平方和\(S(n)=\sum_{\substack{r=1 \\ (r,n)=1}}^{\lfloor n/e \rfloor}\frac{1}{r^2}\)，已知其模\(n\)同余形式。本文将结果扩展到某些奇数次互反和，建立了\(S_m(n)=\sum_{\substack{r=1 \\ (r,n)=1}}^{\lfloor n/e \rfloor}\frac{1}{r^m}\)模\(n\)的统一同余。还研究了广义莱默型乘积\(\prod_{d \mid n}\binom{kd-1}{\lfloor d/e \rfloor}^{\mu(n/d)}\)，虽已有模\(n^3\)同余，但高阶同余无简单闭式，为此通过完全指数贝尔多项式得到显式截断展开。结果为涉及互反和及相关乘积表达式的高阶同余的显式计算和算法验证提供了统一框架。
英文摘要：
This paper investigates high-order congruences of reciprocal power sums and Lehmer-type products. Let $n\geq 1$ with $(n,6)=1$ and $e\in\{2,3,4,6\}$. For the reciprocal square sums \begin{equation*} S(n)=\sum_{\substack{r=1 \\ (r,n)=1}}^{\lfloor n/e \rfloor}\frac{1}{r^2} \end{equation*} we already know the form of the congruence modulo $n$. In this paper, motivated by the known congruences, we first extend these results to certain reciprocal sums of odd order and establish a uniform congruence modulo $n$ for \begin{equation*} S_m(n)=\sum_{\substack{r=1 \\ (r,n)=1}}^{\lfloor n/e \rfloor}\frac{1}{r^m} \end{equation*} We then study the generalized Lehmer-type product \begin{equation*} \prod_{d \mid n}\binom{kd-1}{\lfloor d/e \rfloor}^{μ(n/d)} \end{equation*} Although congruences modulo $n^3$ for this product have previously been obtained, higher-order congruences do not admit a comparably simple closed form. To address this difficulty, we derive an explicit truncated expansion in terms of complete exponential Bell polynomials. The results provide a unified framework for explicit computation and algorithmic verification of higher-order congruences involving reciprocal sums and related product expressions.
