A Family of Supercongruences Involving Multiple Harmonic Sums

TL;DR

This paper establishes a broad family of supercongruences involving multiple harmonic sums modulo prime powers, extending previous results by considering sums with indices coprime to the prime and summing to multiples of prime powers.

Contribution

It generalizes existing supercongruences for multiple harmonic sums to higher powers of primes with coprimality conditions, expanding the theoretical framework.

Findings

Proved supercongruences modulo prime powers for sums with indices coprime to p

Extended previous harmonic sum congruences to more general settings

Established new relations involving multiple harmonic sums and prime powers

Abstract

In recent years, the congruence $$ \sum_{\substack{i+j+k=p\\ i,j,k>0}} \frac1{ijk} \equiv -2 B_{p-3} \pmod{p}, $$ first discovered by the last author have been generalized by either increasing the number of indices and considering the corresponding supercongruences, or by considering the alternating version of multiple harmonic sums. In this paper, we prove a family of similar supercongruences modulo prime powers $p^r$ with the indexes summing up to $mp^r$ where $m$ is coprime to $p$, where all the indexes are also coprime to $p$.