On a kind of generilized multi-harmonic sum

TL;DR

This paper generalizes a known prime modulus congruence involving harmonic sums and Bernoulli numbers using congruence theory and combinatorial methods, revealing new interesting congruences.

Contribution

It introduces a generalized approach to harmonic sum congruences, extending prior results with novel congruences derived through combinatorial and congruence techniques.

Findings

Derived new congruences involving harmonic sums and Bernoulli numbers.

Extended known prime modulus congruences to broader classes of sums.

Applied combinatorial methods to establish these generalized congruences.

Abstract

Let $p$ be an odd prime, Jianqiang Zhao has established a curious congruence $$ \sum_{i+j+k=p \atop i,j,k > 0} \frac{1}{ijk} \equiv -2B_{p-3}\pmod p , $$ where $B_{n}$ denotes the $n-$th Bernoulli numbers. In this paper, we will generalize this problem by using congruent theory and combinatorial methods, and we get some curious congruences.