Abstract
In 2014, Wang and Cai established the following harmonic congruence for any odd prime p and positive integer r,
$$\sum_{\begin{subarray}{c}i+j+k=p^{r}\\
i,j,k\in\mathcal{P}_{p}\end{subarray}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} \quad\quad(\text{mod} \,\, {p^{r}}),$$
where
$ \mathcal{P}_{n} $ denote the set of positive integers which are prime to n.
In this note, we obtain the congruences for distinct odd primes p, q and positive integers α, β,
$$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\
i,j,k\in\mathcal{P}_{2pq}\end{subarray}}\frac{1}{ijk}\equiv\frac{7}{8}\left(2-%
q\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}\pmod{p^{%
\alpha}} $$
and
$$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\
i,j,k\in\mathcal{P}_{pq}\end{subarray}}\frac{(-1)^{i}}{ijk}\equiv\frac{1}{2}%
\left(q-2\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}%
\pmod{p^{\alpha}}. $$
Elementary methods in the theory of L-functions, VI. On the least prime quadratic residue (mod p)
