On some congruences using multiple harmonic sums of length three and four

TL;DR

This paper investigates specific congruences involving multiple harmonic sums of lengths three and four, providing new results modulo prime powers and confirming conjectures by Z.-W Sun.

Contribution

It determines new congruences for sums involving multiple harmonic sums of lengths three and four modulo p and p^2, and confirms conjectures by Sun.

Findings

Derived new congruences for harmonic sum sums modulo p and p^2.

Confirmed three conjectures by Z.-W Sun related to harmonic sums.

Extended the understanding of harmonic sums of lengths three and four.

Abstract

In the present paper, we determine the sums $\sum_{j=1}^{p-1}\frac{H_j^{(s_1)}H_j^{(s_3)}}{j^{s_2}}$ and $\sum_{j=1}^{p-1}\frac{H_j^{(s_1)}H_j^{(s_3)}H_j^{(s_4)}}{j^{s_2}}$ modulo $p$ and modulo $p^2$ in certain cases. This is done by using multiple harmonic sums of length three and four, as well as, many other results. In addition, We recover three congruences conjectured by Z.-W Sun and solved later by the author himself and R. Me\v{s}trovi\'c.