On a kind of generalized multi-harmonic sums

TL;DR

This paper generalizes a known congruence involving harmonic sums and Bernoulli numbers to sums modulo prime powers, expanding the understanding of such congruences in number theory.

Contribution

It introduces a broader class of harmonic sum congruences modulo prime powers, extending previous results by Jianqiang Zhao.

Findings

Established new congruences for generalized harmonic sums modulo prime powers

Extended Zhao's congruence to sums involving higher powers of primes

Provided proofs for a family of similar congruences in number theory

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 $p^r$.