On a congruence involving harmonic series and Bernoulli numbers

Abstract

In 2003, Zhao discovered a curious congruence involving harmonic series and Bernoulli numbers: for any odd prime $p$, $$\sum_{\substack{i,j,k\ge 1\\\gcd(ijk,p)=1\ı+j+k=p}}\frac{1}{ijk}\equiv -2B_{p-3} \pmod{p},$$ where $B_n$ is the $n$-th Bernoulli number. This congruence was generalized by Wang and Cai in 2014, and Cai, Shen and Jia in 2017 by replacing the odd prime $p$ in the summation and modulus with an odd prime power, and a product of two odd prime powers, respectively. In particular, Cai, Shen and Jia proposed a conjectural congruence: for any positive integer $n$ with an odd prime factor $p$ such that $p^r\parallel n$ where $r\ge 1$, $$\sum_{\substack{i,j,k\ge 1\\\gcd(ijk,n)=1\ı+j+k=n}}\frac{1}{ijk}\equiv -2B_{p-3}\cdot \frac{n}{p}\cdot \prod_{\substack{\text{prime $q\mid n$}\\q\ne p}}\left(1-\frac{2}{q}\right)\left(1-\frac{1}{q^3}\right) \pmod{p^r}.$$ In this paper, we establish the following generalization of their conjecture: for any positive integer $n$ with an odd prime factor $p$ such that $p^r\parallel n$ where $r\ge 1$, $$\begin{array}{rl}\sum _{\begin{array}{c}i,j,k\ge 1\\ \gcd (ijk,n)=1\\ a_{1}i+a_{2}j+a_{3}k=An\end{array}}\frac{1}{ijk}& \equiv -2B_{p-3}\cdot \frac{n}{p}\cdot \frac{A{g}^3}{3}\left(\frac{1}{a_1^2{g}_1^2}+\frac{1}{a_2^2{g}_2^2}+\frac{1}{a_3^2{g}_3^2}\right)\\ & \times \prod _{\begin{array}{c}\text{prime q∣n}\\ q\ne p\end{array}}\left(1-\frac{2}{q}\right)\left(1-\frac{1}{q^3}\right)(\mathrm{mod}{p}^r),\end{array}$$ where $a_1$, $a_2$ and $a_3$ are positive integers coprime to $p$, and $A$ is a positive common multiple of $a_1$, $a_2$ and $a_3$. Also, $g_1=\gcd (a_2,a_3)$, $g_2=\gcd (a_3,a_1)$, $g_3=\gcd (a_1,a_2)$ and $g=\gcd (a_1,a_2,a_3)$.