A congruence involving harmonic sums modulo $p^{\alpha}q^{\beta}$

TL;DR

This paper extends harmonic sum congruences to products of two odd prime powers, providing new formulas and conditions for divisibility, and proposes a conjecture for general odd composite numbers involving harmonic sums and Bernoulli numbers.

Contribution

It introduces a new congruence for harmonic sums modulo products of two odd prime powers and establishes conditions for divisibility, extending previous results.

Findings

Derived a congruence for Z(p^α q^β) modulo p^α.

Established necessary and sufficient conditions for Z(p^α q^β) to be divisible by p^α q^β.

Proposed a conjecture for harmonic sums modulo composite numbers involving Bernoulli numbers.

Abstract

In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \begin{equation*} Z(p^{r})\equiv-2p^{r-1}B_{p-3} ~(\bmod ~ p^{r}), \end{equation*} where $ Z(n)=\sum\limits_{i+j+k=n\atop{i,j,k\in\mathcal{P}_{n}}}\frac{1}{ijk}$ and $\mathcal{P}_{n}$ denote the set of positive integers which are prime to $n$. In this note, we obtain a congruence for distinct odd primes $p,~q$ and positive integers $\alpha,~\beta$, \begin{equation*} Z(p^{\alpha}q^{\beta})\equiv 2(2-q)(1-\frac{1}{q^{3}})p^{\alpha-1}q^{\beta-1}B_{p-3}\pmod{p^{\alpha}} \end{equation*} and the necessary and sufficient condition for \begin{equation*} Z(p^{\alpha}q^{\beta})\equiv 0\pmod{p^{\alpha}q^{\beta}}. \end{equation*} Finally, we raise a conjecture that for $n>1$ and odd prime power $p^{\alpha}||n$, $\alpha\geq1$, \begin{eqnarray} \nonumber Z(n)\equiv \prod\limits_{q|n\atop{q\neq p}}(1-\frac{2}{q})(1-\frac{1}{q^{3}})(-\frac{2n}{p})B_{p-3}\pmod{p^{\alpha}}. \end{eqnarray}