New Congruences on Multiple Harmonic Sums and Bernoulli Numbers

Abstract

Let ${\mathcal{P}_{n}}$ denote the set of positive integers which are prime to $n$. Let $B_{n}$ be the $n$-th Bernoulli number. For any prime $p \ge 11$ and integer $r\ge 2$, we prove that $$ \sum\limits_{\begin{smallmatrix} {{l}_{1}}+{{l}_{2}}+\cdots +{{l}_{6}}={{p}^{r}} {{l}_{1}},\cdots ,{{l}_{6}}\in {\mathcal{P}_{p}} \end{smallmatrix}}{\frac{1}{{{l}_{1}}{{l}_{2}}{{l}_{3}}{{l}_{4}}{{l}_{5}}{l}_{6}}}\equiv - \frac{{5!}}{18}p^{r-1}B_{p-3}^{2} \pmod{{{p}^{r}}}. $$ This extends a family of curious congruences. We also obtain other interesting congruences involving multiple harmonic sums and Bernoulli numbers.