On a kind of generalized multi-harmonic sums
Jiaqi Wang, Rong Ma
TL;DR
The paper addresses generalizing Zhao's congruence for the triple sum 1/(ijk) with i+j+k=p to sums Z(p1...pn) and Z(p1^{r1}...pn^{rn}) where indices are coprime to the modulus. It introduces Z(n) and auxiliary sums S and T, and derives a family of congruences modulo prime powers p^r using inclusion-exclusion and a suite of lemmas on generalized harmonic sums, linking results to Bernoulli numbers B_{p1-3}. The main contributions are explicit congruences for Z(p1...pn) and Z(p1^{r1}...pn^{rn}) in terms of B_{p1-3} and correction terms dependent on the other primes, with corollaries recapturing known results for n=1,2. This work extends the landscape of generalized multi-harmonic sum congruences and provides a structured framework for evaluating similar sums modulo prime powers.
Abstract
Let $p$ be an odd prime, Jianqiang Zhao has established a curious congruence, which is $$ \sum_{i+j+k=p \atop i,j,k > 0} \frac{1}{ijk} \equiv -2B_{p-3}\pmod p , $$ where $B_{n}$ denotes the $n$-th Bernoulli number. In this paper, we will generalize this kind of sums and prove a family of similar congruences modulo prime powers $p^r$.