Congruences Involving Multiple Harmonic Sums and Finite Multiple Zeta Values

TL;DR

This paper extends known harmonic sum congruences involving Bernoulli numbers to sums with more variables, connecting them to finite multiple zeta values and providing new modular identities.

Contribution

It generalizes harmonic sum congruences to multiple variables and links these to finite multiple zeta values generated by Bernoulli numbers.

Findings

Proved congruences for sums with n=2 or 4 variables involving Bernoulli numbers.

Extended harmonic sum identities to higher dimensions with prime power moduli.

Connected multiple harmonic sums to 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 $p>2$ $$ \sum_{\substack{i+j+k=p^r\\ i,j,k\in{\mathfrak P}_p}} \frac1{ijk} \equiv -2p^{r-1} B_{p-3} \pmod{p^r}, $$ 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.