## Arithmetic theory of harmonic numbers

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**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.## References

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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
- 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