Arithmetic theory of harmonic numbers

Author:
Zhi-Wei Sun

Journal:
Proc. Amer. Math. Soc. **140** (2012), 415-428

MSC (2010):
Primary 11B75; Secondary 05A19, 11A07, 11B68

DOI:
https://doi.org/10.1090/S0002-9939-2011-10925-0

Published electronically:
June 8, 2011

MathSciNet review:
2846311

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Harmonic numbers $H_{k}=\sum _{0<j\leqslant k}1/j\ (k=0,1,2,\ldots )$ play important roles in mathematics. In this paper we investigate their arithmetic properties and obtain various basic congruences. Let $p>3$ be a prime. We show that \begin{equation*} \sum _{k=1}^{p-1}\frac {H_{k}}{k2^{k}}\equiv 0\ (\mathrm {mod} \ p),\ \sum _{k=1}^{p-1}H_{k}^{2} \equiv 2p-2\ (\mathrm {mod} \ p^{2}), \ \sum _{k=1}^{p-1}H_{k}^{3}\equiv 6\ (\mathrm {mod} \ p),\end{equation*} and \begin{equation*} \sum _{k=1}^{p-1}\frac {H_{k}^{2}}{k^{2}}\equiv 0\ (\mathrm {mod} \ p)\qquad \text {provided }\ p>5. \end{equation*} (In contrast, it is known that $\sum _{k=1}^{\infty }H_{k}/(k2^{k})=\pi ^{2}/12$ and $\sum _{k=1}^{\infty }H_{k}^{2}/k^{2}=17\pi ^{4}/360$.) Our tools include some sophisticated combinatorial identities and properties of Bernoulli numbers.

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Additional Information

**Zhi-Wei Sun**

Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China

MR Author ID:
254588

Email:
zwsun@nju.edu.cn

Keywords:
Harmonic numbers,
congruences,
Bernoulli numbers

Received by editor(s):
July 22, 2010

Received by editor(s) in revised form:
November 23, 2010

Published electronically:
June 8, 2011

Additional Notes:
The author was supported by the National Natural Science Foundation (grant 10871087) of China

Communicated by:
Matthew A. Papanikolas

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.