Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Published electronically: June 8, 2011
MathSciNet review: 2846311
Full-text PDF

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

$\displaystyle \sum _{k=1}^{p-1}\frac{H_{k}}{k2^{k}}\equiv 0 (\mathrm{mod} p),\ ... ...2p-2 (\mathrm{mod} p^{2}), \sum _{k=1}^{p-1}H_{k}^{3}\equiv 6 (\mathrm{mod} p),$


$\displaystyle \sum _{k=1}^{p-1}\frac {H_{k}^{2}}{k^{2}}\equiv 0 (\mathrm{mod} \ p)$$\displaystyle \qquad \text {provided } p>5. $

(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 [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11B75, 05A19, 11A07, 11B68

Retrieve articles in all journals with MSC (2010): 11B75, 05A19, 11A07, 11B68

Additional Information

Zhi-Wei Sun
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China

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.

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia