Arithmetic theory of harmonic numbers (II)

Abstract

For $k=1,2,\dots$ let $H_k$ denote the harmonic number ${\sum }_{j=1}^{k}1/j$. In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime $p>3$ we have $$\sum _{k=1}^{p-1}\frac{H_k}{k{2}^k}\equiv \frac{7}{24}p{B}_{p-3}(\mathrm{mod}{p}^2),\sum _{k=1}^{p-1}\frac{H_{k,2}}{k{2}^k}\equiv -\frac{3}{8}{B}_{p-3}(\mathrm{mod}p),$$ and $$\sum _{k=1}^{p-1}\frac{H_{k,2n}^2}{k^{2n}}\equiv \frac{\left(\genfrac{}{}{0ex}{}{6n+1}{2n-1}\right)+n}{6n+1}p{B}_{p-1-6n}(\mathrm{mod}{p}^2)$$ for any positive integer $n<(p-1)/6$, where $B_0,B_1,B_2,\dots$ are Bernoulli numbers, and $H_{k,m}:={\sum }_{j=1}^{k}1/j^m$.

13 pages. Final published version

Cited by in corpus (3)

• Open Conjectures on Congruences
• Proof of a congruence for harmonic numbers conjectured by Z.-W. Sun
• An Extension of a Congruence by Kohnen