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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
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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.