# Congruences Involving Multiple Harmonic Sums and Finite Multiple Zeta
Values

Research paper by **Jianqiang Zhao**

Indexed on: **08 Jan '16**Published on: **08 Jan '16**Published in: **Mathematics - Number Theory**

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#### Abstract

Let $p$ be a prime and ${\mathfrak P}_p$ the set of positive integers which
are prime to $p$. Recently, Wang and Cai proved that for every positive integer
$r$ and prime $p>2$ $$ \sum_{\substack{i+j+k=p^r\\ i,j,k\in{\mathfrak P}_p}}
\frac1{ijk} \equiv
-2p^{r-1} B_{p-3} \pmod{p^r}, $$ where $B_{p-3}$ is the $(p-3)$-rd Bernoulli
number. In this paper we prove the following analogous result: Let $n=2$ or
$4$. Then for every positive integer $r\ge n/2$ and prime $p>4$ $$
\sum_{\substack{i_1+\cdots+i_n=p^r\\ i_1,\dots,i_n\in{\mathfrak P}_p}}
\frac1{i_1i_2\cdots i_n} \equiv
-\frac{n!}{n+1} p^{r} B_{p-n-1} \pmod{p^{r+1}}. $$ Moreover, by using integer
relation detecting tool PSLQ we can show that generalizations with larger
integers $n$ should involving finite multiple zeta values generated by
Bernoulli numbers.