On a congruence involving harmonic series and Bernoulli numbers

TL;DR

This paper generalizes a known congruence involving harmonic series and Bernoulli numbers, extending it to more complex sums with linear constraints and multiple parameters, revealing deeper number-theoretic relationships.

Contribution

It establishes a broad generalization of previous conjectures, incorporating linear combinations and gcd conditions in harmonic sum congruences involving Bernoulli numbers.

Findings

Proves a new congruence involving sums with linear constraints and gcd conditions.

Extends previous conjectures to more general settings with multiple parameters.

Provides explicit formulas connecting harmonic sums, Bernoulli numbers, and prime factorizations.

Abstract

In 2003, Zhao discovered a curious congruence involving harmonic series and Bernoulli numbers: for any odd prime $p$, $$\sum_{\substack{i,j,k\ge 1\\\gcd(ijk,p)=1\\i+j+k=p}}\frac{1}{ijk}\equiv -2B_{p-3} \pmod{p},$$ 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 $r\ge 1$, $$\sum_{\substack{i,j,k\ge 1\\\gcd(ijk,n)=1\\i+j+k=n}}\frac{1}{ijk}\equiv -2B_{p-3}\cdot \frac{n}{p}\cdot \prod_{\substack{\text{prime $q\mid n$}\\q\ne p}}\left(1-\frac{2}{q}\right)\left(1-\frac{1}{q^3}\right) \pmod{p^r}.$$ 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 $r\ge 1$, $$\begin{aligned} \sum_{\substack{i,j,k\ge 1\\\gcd(ijk,n)=1\\a_1 i+a_2 j+a_3 k=An}}\frac{1}{ijk}&\equiv -2B_{p-3}\cdot \frac{n}{p}\cdot \frac{Ag^3}{3}\left(\frac{1}{a_1^2 g_1^2}+\frac{1}{a_2^2 g_2^2}+\frac{1}{a_3^2 g_3^2}\right)\\ &\quad\times \prod_{\substack{\text{prime $q\mid n$}\\q\ne p}}\left(1-\frac{2}{q}\right)\left(1-\frac{1}{q^3}\right) \pmod{p^r}, \end{aligned}$$ 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 $g=\gcd(a_1,a_2,a_3)$.