On the mod $p^2$ determination of $\sum_{k=1}^{p-1}H_k/(k\cdot 2^k)$: another proof of a conjecture by Sun

TL;DR

This paper provides an alternative proof for Sun's conjecture on a harmonic sum modulo p^2, using new congruences for related sums involving harmonic numbers and powers of 2.

Contribution

It introduces novel congruences for sums involving harmonic numbers and powers of 2, offering a different proof of Sun's conjecture.

Findings

Confirmed Sun's conjecture modulo p^2.

Established congruences for sums of harmonic numbers with powers of 2.

Provided methods applicable to related harmonic sum congruences.

Abstract

For a positive integer $n$ let $H_n=\sum_{k=1}^{n}1/k$ be the $n$th harmonic number. Z. W. Sun conjectured that for any prime $p\ge 5$, $$ \sum_{k=1}^{p-1}\frac{H_k}{k\cdot 2^k} \equiv7/24pB_{p-3}\pmod{p^2}. $$ This conjecture is recently confirmed by Z. W. Sun and L. L. Zhao. In this note we give another proof of the above congruence by establishing congruences for all the sums of the form $\sum_{k=1}^{p-1}2^{\pm k}H_k^r/k^s \,(\bmod{\, p^{4-r-s}})$ with $(r,s)\in\{(1,1),(1,2),(2,1) \}$.