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

TL;DR

This paper extends harmonic sum congruences involving primes to composite numbers with multiple prime factors, providing new formulas and conjectures relating these sums to Bernoulli numbers.

Contribution

It introduces new congruences for alternating and non-alternating harmonic sums modulo prime powers involving two distinct odd primes and proposes a general conjecture for composite numbers.

Findings

Derived explicit congruences for sums involving two primes and their powers.

Established relationships between harmonic sums and Bernoulli numbers.

Proposed a conjecture for harmonic sums over composite numbers with multiple prime factors.

Abstract

In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \begin{equation*} \sum\limits_{i+j+k=p^{r}\atop{i,j,k\in \mathcal{P}_{p}}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} ~(\bmod ~ p^{r}), \end{equation*} 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 $\alpha,~\beta$, \begin{equation*} \sum\limits_{i+j+k=p^{\alpha}q^{\beta}\atop{i,j,k\in\mathcal{P}_{pq}\atop{i\equiv j\equiv k\equiv 1\pmod{2}}}}\frac{1}{ijk}\equiv\frac{7}{8}(2-q)(1-\frac{1}{q^{3}})p^{\alpha-1}q^{\beta-1}B_{p-3}\pmod{p^{\alpha}} \end{equation*} and \begin{equation*} \sum\limits_{i+j+k=p^{\alpha}q^{\beta}\atop{i,j,k\in \mathcal{P}_{pq}}}\frac{(-1)^{i}}{ijk} \equiv \frac{1}{2}(q-2)(1-\frac{1}{q^{3}})p^{\alpha-1}q^{\beta-1}B_{p-3}\pmod{p^{\alpha}}. \end{equation*} Finally, we raise a conjecture that for $n>1$ and odd prime power $p^{\alpha}||n$, $\alpha\geq1$, \begin{eqnarray} \nonumber \sum\limits_{i+j+k=n\atop{i,j,k\in\mathcal{P}_{n}}}\frac{(-1)^{i}}{ijk} \equiv \prod\limits_{q|n\atop{q\neq p}}(1-\frac{2}{q})(1-\frac{1}{q^{3}})\frac{n}{2p}B_{p-3}\pmod{p^{\alpha}} \end{eqnarray} and \begin{eqnarray} \nonumber \sum\limits_{i+j+k=n\atop{i,j,k\in\mathcal{P}_{n}\atop{i\equiv j\equiv k\equiv 1\pmod{2}}}}\frac{1}{ijk} \equiv \prod\limits_{q|n\atop{q\neq p}}(1-\frac{2}{q})(1-\frac{1}{q^{3}})(-\frac{7n}{8p})B_{p-3}\pmod{p^{\alpha}}. \end{eqnarray}