Arithmetic theory of harmonic numbers (II)

Zhi-Wei Sun¹ ; Li-Lu Zhao²

¹ Department of Mathematics Nanjing University Nanjing 210093, People's Republic of China
² School of Mathematics Hefei University of Technology Hefei 230009, People's Republic of China

Colloquium Mathematicum, Tome 130 (2013) no. 1, pp. 67-78

Résumé

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 $H_{k,m}:=\sum_{j=1}^k 1/j^m$.

Détail

DOI : 10.4064/cm130-1-7

Keywords: ldots denote harmonic number sum paper establish congruences involving harmonic numbers example prime have def frac sum p equiv p pmod quad sum p equiv p pmod sum p equiv binom n p pmod positive integer p where ldots bernoulli numbers sum