Arithmetic theory of harmonic numbers (II)
Volume 130 / 2013
Colloquium Mathematicum 130 (2013), 67-78
MSC: Primary 11A07, 11B68;
Secondary 05A19, 11B75.
DOI: 10.4064/cm130-1-7
Abstract
For $k=1,2,\ldots$ 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 $$\def\f#1#2{\frac{#1}{#2}}\sum_{k=1}^{p-1}\f{H_k}{k2^k}\equiv\f7{24}pB_{p-3}\pmod{p^2},\quad \ \sum_{k=1}^{p-1}\f{H_{k,2}}{k2^k}\equiv-\f 38B_{p-3}\pmod{p},$$ and $$\sum_{k=1}^{p-1}\f{H_{k,2n}^2}{k^{2n}}\equiv\f{\binom{6n+1}{2n-1}+n}{6n+1}pB_{p-1-6n} \pmod{p^2}$$ for any positive integer $n<(p-1)/6$, where $B_0,B_1,B_2,\ldots$ are Bernoulli numbers, and $H_{k,m}:=\sum_{j=1}^k 1/j^m$.
