Congruences Involving Multiple Harmonic Sums and Finite Multiple Zeta Values

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 $p>2$ $$\sum _{\begin{array}{c}i+j+k=p^r\\ i,j,k\in {\mathfrak{P}}_p\end{array}}\frac{1}{ijk}\equiv -2p^{r-1}{B}_{p-3}(\mathrm{mod}{p}^r),$$ 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 $p>4$ $$\sum _{\begin{array}{c}{i}_1+\cdots +i_n=p^r\\ i_1,\dots ,i_n\in {\mathfrak{P}}_p\end{array}}\frac{1}{i_1{i}_2\cdots i_n}\equiv -\frac{n!}{n+1}{p}^r{B}_{p-n-1}(\mathrm{mod}{p}^{r+1}).$$ 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.

13 pages, we added a newly discovered connection to the finite multiple zeta values in the last section

References in corpus (1)

• New Congruences on Multiple Harmonic Sums and Bernoulli Numbers

Cited by in corpus (2)

• Congruences related to Miki's identity
• Some Families of Super Congruences Involving Alternating Multiple Harmonic Sums