Bounds of divided universal Bernoulli numbers and universal Kummer congruences
HTML articles powered by AMS MathViewer
- by Arnold Adelberg, Shaofang Hong and Wenli Ren
- Proc. Amer. Math. Soc. 136 (2008), 61-71
- DOI: https://doi.org/10.1090/S0002-9939-07-09025-9
- Published electronically: August 14, 2007
- PDF | Request permission
Abstract:
Let $p$ be a prime. We obtain good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number $\tfrac {\hat {B}_n}{n}$ when $n$ is divisible by $p-1$. As an application, we give a simple proof of Clarke’s 1989 universal von Staudt theorem. We also establish the universal Kummer congruences modulo $p$ for the divided universal Bernoulli numbers for the case $(p-1)|n$, which is a new result.References
- Arnold Adelberg, Universal higher order Bernoulli numbers and Kummer and related congruences, J. Number Theory 84 (2000), no. 1, 119–135. MR 1782265, DOI 10.1006/jnth.2000.2526
- Arnold Adelberg, Kummer congruences for universal Bernoulli numbers and related congruences for poly-Bernoulli numbers, Int. Math. J. 1 (2002), no. 1, 53–63. MR 1825492
- Arnold Adelberg, Universal Kummer congruences mod prime powers, J. Number Theory 109 (2004), no. 2, 362–378. MR 2106486, DOI 10.1016/j.jnt.2004.07.005
- Francis Clarke, The universal von Staudt theorems, Trans. Amer. Math. Soc. 315 (1989), no. 2, 591–603. MR 986687, DOI 10.1090/S0002-9947-1989-0986687-3
- Francis Clarke and Christine Jones, A congruence for factorials, Bull. London Math. Soc. 36 (2004), no. 4, 553–558. MR 2069019, DOI 10.1112/S0024609304003194
- I. Dibag, An analogue of the von Staudt-Clausen theorem, J. Algebra 87 (1984), no. 2, 332–341. MR 739937, DOI 10.1016/0021-8693(84)90140-6
- Shaofang Hong, Notes on Glaisher’s congruences, Chinese Ann. Math. Ser. B 21 (2000), no. 1, 33–38. MR 1762270, DOI 10.1142/S0252959900000054
- A. Hurwitz, Über die Entwicklungskoeffizienten der lemniskatischen Funktionen, Math. Ann. 51 (1899), 196-226.
- Nicholas M. Katz, The congruences of Clausen-von Staudt and Kummer for Bernoulli-Hurwitz numbers, Math. Ann. 216 (1975), 1–4. MR 387293, DOI 10.1007/BF02547966
- M. Ram Murty, Introduction to $p$-adic analytic number theory, AMS/IP Studies in Advanced Mathematics, vol. 27, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002. MR 1913413, DOI 10.1090/amsip/027
- Nigel Ray, Extensions of umbral calculus: penumbral coalgebras and generalised Bernoulli numbers, Adv. in Math. 61 (1986), no. 1, 49–100. MR 847728, DOI 10.1016/0001-8708(86)90065-4
- Wen-li Ren, Shao-fang Hong, and Xing-wang Zhou, A generalization of Wilson’s theorem, Sichuan Daxue Xuebao 43 (2006), no. 3, 517–519 (English, with English and Chinese summaries). MR 2241947
- Lawrence C. Washington, $p$-adic $L$-functions and sums of powers, J. Number Theory 69 (1998), no. 1, 50–61. MR 1611077, DOI 10.1006/jnth.1997.2195
Bibliographic Information
- Arnold Adelberg
- Affiliation: Department of Mathematics, Grinnell College, Grinnell, Iowa 50112-0806
- Email: adelbe@math.grinnell.edu
- Shaofang Hong
- Affiliation: Mathematical College, Sichuan University, Chengdu 610064, People’s Republic of China
- Email: s-f.hong@tom.com, hongsf02@yahoo.com
- Wenli Ren
- Affiliation: Mathematical College, Sichuan University, Chengdu 610064, People’s Republic of China, and Department of Mathematics, Dezhou University, Dezhou 253023, People’s Republic of China
- Email: renwenli80@163.com
- Received by editor(s): July 5, 2006
- Received by editor(s) in revised form: December 1, 2006
- Published electronically: August 14, 2007
- Additional Notes: The second author is the corresponding author and was supported by New Century Excellent Talents in University Grant # NCET-06-0785, and by SRF for ROCS, SEM
- Communicated by: Wen-Ching Winnie Li
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 61-71
- MSC (2000): Primary 11B68, 11B83; Secondary 11A07
- DOI: https://doi.org/10.1090/S0002-9939-07-09025-9
- MathSciNet review: 2350389