# Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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## Arithmetic theory of harmonic numbersHTML articles powered by AMS MathViewer

by Zhi-Wei Sun
Proc. Amer. Math. Soc. 140 (2012), 415-428 Request permission

## 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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• 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
• 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
• © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
• 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
• MathSciNet review: 2846311