Higher-Order Congruence for Reciprocal Power Sums and Generalized Lehmer-Type Products

Abstract

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.