Arithmetic theory of harmonic numbers
Author:
ZhiWei Sun
Journal:
Proc. Amer. Math. Soc. 140 (2012), 415428
MSC (2010):
Primary 11B75; Secondary 05A19, 11A07, 11B68
Published electronically:
June 8, 2011
MathSciNet review:
2846311
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Abstract: Harmonic numbers play important roles in mathematics. In this paper we investigate their arithmetic properties and obtain various basic congruences. Let be a prime. We show that and (In contrast, it is known that and .) Our tools include some sophisticated combinatorial identities and properties of Bernoulli numbers.
 [A]
Emre
Alkan, Variations on Wolstenholme’s theorem, Amer. Math.
Monthly 101 (1994), no. 10, 1001–1004. MR 1304326
(95g:11001), 10.2307/2975168
 [ASVZ]
E.
Alkan, J.
Sneed, M.
Vâjâitu, and A.
Zaharescu, Wolstenholme matrices, Math. Rep. (Bucur.)
8(58) (2006), no. 1, 1–8. MR 2268379
(2007f:11002)
 [B]
M.
Bayat, A generalization of Wolstenholme’s theorem, Amer.
Math. Monthly 104 (1997), no. 6, 557–560. MR 1453658
(98e:11007), 10.2307/2975083
 [BPQ]
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
 [BB]
David
Borwein and Jonathan
M. Borwein, On an intriguing integral and some
series related to 𝜁(4), Proc. Amer.
Math. Soc. 123 (1995), no. 4, 1191–1198. MR 1231029
(95e:11137), 10.1090/S0002993919951231029X
 [C]
S. W. Coffman, Problem 1240 and Solution: An infinite series with harmonic numbers, Math. Mag. 60 (1987), 118119.
 [FPZ]
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
(2010j:11039), 10.1016/j.jnt.2009.05.018
 [G]
Henry
W. Gould, Combinatorial identities, Henry W. Gould,
Morgantown, W.Va., 1972. A standardized set of tables listing 500 binomial
coefficient summations. MR 0354401
(50 #6879)
 [Gr]
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
(99h:11016)
 [HT]
Charles
Helou and Guy
Terjanian, On Wolstenholme’s theorem and its converse,
J. Number Theory 128 (2008), no. 3, 475–499. MR 2389852
(2008k:11003), 10.1016/j.jnt.2007.06.008
 [IR]
Kenneth
Ireland and Michael
Rosen, A classical introduction to modern number theory, 2nd
ed., Graduate Texts in Mathematics, vol. 84, SpringerVerlag, New
York, 1990. MR
1070716 (92e:11001)
 [L]
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, 10.2307/1968791
 [PS]
Hao
Pan and ZhiWei
Sun, New identities involving Bernoulli and Euler polynomials,
J. Combin. Theory Ser. A 113 (2006), no. 1,
156–175. MR 2192774
(2006j:05020), 10.1016/j.jcta.2005.07.008
 [St]
Richard
P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge
Studies in Advanced Mathematics, vol. 49, Cambridge University Press,
Cambridge, 1997. With a foreword by GianCarlo Rota; Corrected reprint of
the 1986 original. MR 1442260
(98a:05001)
 [SP]
ZhiWei
Sun and Hao
Pan, Identities concerning Bernoulli and Euler polynomials,
Acta Arith. 125 (2006), no. 1, 21–39. MR 2275215
(2007i:11037), 10.4064/aa12513
 [ST]
ZhiWei
Sun and Roberto
Tauraso, New congruences for central binomial coefficients,
Adv. in Appl. Math. 45 (2010), no. 1, 125–148.
MR
2628791 (2011i:11025), 10.1016/j.aam.2010.01.001
 [W]
J. Wolstenholme, On certain properties of prime numbers, Quart. J. Math. 5 (1862), 3539.
 [Z]
Jianqiang
Zhao, Wolstenholme type theorem for multiple harmonic sums,
Int. J. Number Theory 4 (2008), no. 1, 73–106.
MR
2387917 (2008m:11006), 10.1142/S1793042108001146
 [A]
 E. Alkan, Variations on Wolstenholme's theorem, Amer. Math. Monthly 101 (1994), 10011004. MR 1304326 (95g:11001)
 [ASVZ]
 E. Alkan, J. Sneed, M. Vajaitu and A. Zaharescu, Wolstenholme matrices, Math. Rep. 58 (2006), 18. MR 2268379 (2007f:11002)
 [B]
 M. Bayat, A generalization of Wolstenholme's theorem, Amer. Math. Monthly 104 (1997), 557560. MR 1453658 (98e:11007)
 [BPQ]
 A. T. Benjamin, G. O. Preston and J. J. Quinn, A Stirling encounter with harmonic numbers, Math. Mag. 75 (2002), 95103. MR 1573592
 [BB]
 D. Borwein and J. M. Borwein, On an intriguing integral and some series related to , Proc. Amer. Math. Soc. 123 (1995), 11911198. MR 1231029 (95e:11137)
 [C]
 S. W. Coffman, Problem 1240 and Solution: An infinite series with harmonic numbers, Math. Mag. 60 (1987), 118119.
 [FPZ]
 A. M. Fu, H. Pan and I. F. Zhang, Symmetric identities on Bernoulli polynomials, J. Number Theory 129 (2009), 26962701. MR 2549525 (2010j:11039)
 [G]
 H. W. Gould, Combinatorial Identities, Morgantown Printing and Binding Co., 1972. MR 0354401 (50:6879)
 [Gr]
 A. Granville, Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers, in: Organic Mathematics (Burnaby, BC, 1995), pp. 253276, CMS Conf. Proc., 20, Amer. Math. Soc., Providence, RI, 1997. MR 1483922 (99h:11016)
 [HT]
 C. Helou and G. Terjanian, On Wolstenholme's theorem and its converse, J. Number Theory 128 (2008), 475499. MR 2389852 (2008k:11003)
 [IR]
 K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory (Graduate Texts in Math., 84), 2nd ed., Springer, New York, 1990. MR 1070716 (92e:11001)
 [L]
 E. Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of Math. 39 (1938), 350360. MR 1503412
 [PS]
 H. Pan and Z. W. Sun, New identities involving Bernoulli and Euler polynomials, J. Combin. Theory Ser. A 113 (2006), 156175. MR 2192774 (2006j:05020)
 [St]
 R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge Univ. Press, Cambridge, 1999. MR 1442260 (98a:05001)
 [SP]
 Z. W. Sun and H. Pan, Identities concerning Bernoulli and Euler polynomials, Acta Arith. 125 (2006), 2139. MR 2275215 (2007i:11037)
 [ST]
 Z. W. Sun and R. Tauraso, New congruences for central binomial coefficients, Adv. in Appl. Math. 45 (2010), 125148. MR 2628791
 [W]
 J. Wolstenholme, On certain properties of prime numbers, Quart. J. Math. 5 (1862), 3539.
 [Z]
 J. Zhao, Wolstenholme type theorem for multiple harmonic sums, Int. J. Number Theory 4 (2008), 73106. MR 2387917 (2008m:11006)
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Additional Information
ZhiWei Sun
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email:
zwsun@nju.edu.cn
DOI:
http://dx.doi.org/10.1090/S000299392011109250
PII:
S 00029939(2011)109250
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.
