Congruences for harmonic sums

Abstract

Zhao found a curious congruence modulo p on harmonic sums. Xia and Cai generalized his congruence to a supercongruence modulo p^2. In this paper, we improve the harmonic sums \[ H_{p}(n)=\sum\limits_{\substack{l_{1}+l_{2}+\cdots+l_{n}=p\\ l_{1}, l_{2}, \ldots , l_{n}>0}} \frac{1}{l_{1} l_{2} \cdots l_{n}} \] to supercongruences modulo p^3 and p^4 for odd and even where prime p>8 and 3 \leq n \leq p-6.