Arithmetic theory of harmonic numbers
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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.References
- Emre Alkan, Variations on Wolstenholme’s theorem, Amer. Math. Monthly 101 (1994), no. 10, 1001–1004. MR 1304326, DOI 10.2307/2975168
- E. Alkan, J. Sneed, M. Vâjâitu, and A. Zaharescu, Wolstenholme matrices, Math. Rep. (Bucur.) 8(58) (2006), no. 1, 1–8. MR 2268379
- M. Bayat, A generalization of Wolstenholme’s theorem, Amer. Math. Monthly 104 (1997), no. 6, 557–560. MR 1453658, DOI 10.2307/2975083
- Arthur T. Benjamin, Gregory O. Preston, and Jennifer J. Quinn, A Stirling Encounter with Harmonic Numbers, Math. Mag. 75 (2002), no. 2, 95–103. MR 1573592, DOI 10.1080/0025570X.2002.11953110
- David Borwein and Jonathan M. Borwein, On an intriguing integral and some series related to $\zeta (4)$, Proc. Amer. Math. Soc. 123 (1995), no. 4, 1191–1198. MR 1231029, DOI 10.1090/S0002-9939-1995-1231029-X
- S. W. Coffman, Problem 1240 and Solution: An infinite series with harmonic numbers, Math. Mag. 60 (1987), 118–119.
- Amy M. Fu, Hao Pan, and Iris F. Zhang, Symmetric identities on Bernoulli polynomials, J. Number Theory 129 (2009), no. 11, 2696–2701. MR 2549525, DOI 10.1016/j.jnt.2009.05.018
- Henry W. Gould, Combinatorial identities, Henry W. Gould, Morgantown, W. Va., 1972. A standardized set of tables listing 500 binomial coefficient summations. MR 0354401
- Andrew Granville, Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers, Organic mathematics (Burnaby, BC, 1995) CMS Conf. Proc., vol. 20, Amer. Math. Soc., Providence, RI, 1997, pp. 253–276. MR 1483922
- Charles Helou and Guy Terjanian, On Wolstenholme’s theorem and its converse, J. Number Theory 128 (2008), no. 3, 475–499. MR 2389852, DOI 10.1016/j.jnt.2007.06.008
- Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716, DOI 10.1007/978-1-4757-2103-4
- Emma Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of Math. (2) 39 (1938), no. 2, 350–360. MR 1503412, DOI 10.2307/1968791
- Hao Pan and Zhi-Wei Sun, New identities involving Bernoulli and Euler polynomials, J. Combin. Theory Ser. A 113 (2006), no. 1, 156–175. MR 2192774, DOI 10.1016/j.jcta.2005.07.008
- Richard P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. With a foreword by Gian-Carlo Rota; Corrected reprint of the 1986 original. MR 1442260, DOI 10.1017/CBO9780511805967
- Zhi-Wei Sun and Hao Pan, Identities concerning Bernoulli and Euler polynomials, Acta Arith. 125 (2006), no. 1, 21–39. MR 2275215, DOI 10.4064/aa125-1-3
- Zhi-Wei Sun and Roberto Tauraso, New congruences for central binomial coefficients, Adv. in Appl. Math. 45 (2010), no. 1, 125–148. MR 2628791, DOI 10.1016/j.aam.2010.01.001
- J. Wolstenholme, On certain properties of prime numbers, Quart. J. Math. 5 (1862), 35–39.
- Jianqiang Zhao, Wolstenholme type theorem for multiple harmonic sums, Int. J. Number Theory 4 (2008), no. 1, 73–106. MR 2387917, DOI 10.1142/S1793042108001146
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