# Mod $p$ structure of alternating and non-alternating multiple harmonic sums

• [1] Department of Mathematics Eckerd College St. Petersburg, Florida 33711, USA
• Volume: 23, Issue: 1, page 299-308
• ISSN: 1246-7405

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## Abstract

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The well-known Wolstenholme’s Theorem says that for every prime $p>3$ the $\left(p\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}1\right)$-st partial sum of the harmonic series is congruent to $0$ modulo ${p}^{2}$. If one replaces the harmonic series by ${\sum }_{k\ge 1}1/{n}^{k}$ for $k$ even, then the modulus has to be changed from ${p}^{2}$ to just $p$. One may consider generalizations of this to multiple harmonic sums (MHS) and alternating multiple harmonic sums (AMHS) which are partial sums of multiple zeta value series and the alternating Euler sums, respectively. A lot of results along this direction have been obtained in the recent articles [6, 7, 8, 10, 11, 12], which we shall summarize in this paper. It turns out that for a prime $p$ the $\left(p-1\right)$-st sum of the general MHS and AMHS modulo $p$ is not congruent to $0$ anymore; however, it often can be expressed by Bernoulli numbers. So it is a quite interesting problem to find out exactly what they are. In this paper we will provide a theoretical framework in which this kind of results can be organized and further investigated. We shall also compute some more MHS modulo a prime $p$ when the weight is less than $13$.

## How to cite

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Zhao, Jianqiang. "Mod $p$ structure of alternating and non-alternating multiple harmonic sums." Journal de Théorie des Nombres de Bordeaux 23.1 (2011): 299-308. <http://eudml.org/doc/219756>.

@article{Zhao2011,
abstract = {The well-known Wolstenholme’s Theorem says that for every prime $p&gt;3$ the $(p\!-\!1)$-st partial sum of the harmonic series is congruent to $0$ modulo $p^2$. If one replaces the harmonic series by $\sum _\{k\ge 1\} 1/n^k$ for $k$ even, then the modulus has to be changed from $p^2$ to just $p$. One may consider generalizations of this to multiple harmonic sums (MHS) and alternating multiple harmonic sums (AMHS) which are partial sums of multiple zeta value series and the alternating Euler sums, respectively. A lot of results along this direction have been obtained in the recent articles [6, 7, 8, 10, 11, 12], which we shall summarize in this paper. It turns out that for a prime $p$ the $(p-1)$-st sum of the general MHS and AMHS modulo $p$ is not congruent to $0$ anymore; however, it often can be expressed by Bernoulli numbers. So it is a quite interesting problem to find out exactly what they are. In this paper we will provide a theoretical framework in which this kind of results can be organized and further investigated. We shall also compute some more MHS modulo a prime $p$ when the weight is less than $13$.},
affiliation = {Department of Mathematics Eckerd College St. Petersburg, Florida 33711, USA},
author = {Zhao, Jianqiang},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Multiple harmonic sums; alternating multiple harmonic sums; duality; shuffle relations; multiple harmonic sums},
language = {eng},
month = {3},
number = {1},
pages = {299-308},
publisher = {Société Arithmétique de Bordeaux},
title = {Mod $p$ structure of alternating and non-alternating multiple harmonic sums},
url = {http://eudml.org/doc/219756},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Zhao, Jianqiang
TI - Mod $p$ structure of alternating and non-alternating multiple harmonic sums
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/3//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 1
SP - 299
EP - 308
AB - The well-known Wolstenholme’s Theorem says that for every prime $p&gt;3$ the $(p\!-\!1)$-st partial sum of the harmonic series is congruent to $0$ modulo $p^2$. If one replaces the harmonic series by $\sum _{k\ge 1} 1/n^k$ for $k$ even, then the modulus has to be changed from $p^2$ to just $p$. One may consider generalizations of this to multiple harmonic sums (MHS) and alternating multiple harmonic sums (AMHS) which are partial sums of multiple zeta value series and the alternating Euler sums, respectively. A lot of results along this direction have been obtained in the recent articles [6, 7, 8, 10, 11, 12], which we shall summarize in this paper. It turns out that for a prime $p$ the $(p-1)$-st sum of the general MHS and AMHS modulo $p$ is not congruent to $0$ anymore; however, it often can be expressed by Bernoulli numbers. So it is a quite interesting problem to find out exactly what they are. In this paper we will provide a theoretical framework in which this kind of results can be organized and further investigated. We shall also compute some more MHS modulo a prime $p$ when the weight is less than $13$.
LA - eng
KW - Multiple harmonic sums; alternating multiple harmonic sums; duality; shuffle relations; multiple harmonic sums
UR - http://eudml.org/doc/219756
ER -

## References

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4. M.E. Hoffman, Quasi-Shuffle Products. J. Algebraic Combin. 11 (2000), 49–68. Zbl0959.16021MR1747062
5. M.E. Hoffman, Algebraic aspects of multiple zeta values. In: Zeta Functions, Topology and Quantum Physics, Developments in Mathematics 14, T. Aoki et. al. (eds.), Springer, 2005, New York, pp. 51–74. arXiv: math.QA/0309425. Zbl1170.11324MR2179272
6. M.E. Hoffman, Quasi-symmetric functions and mod $p$ multiple harmonic sums. arXiv: math.NT/0401319
7. R. Tauraso, Congruences involving alternating multiple harmonic sum. Elec. J. Combinatorics, 17(1) (2010), R16. Zbl1222.11006MR2587747
8. R. Tauraso and J. Zhao, Congruences of alternating multiple harmonic sums. In press: J. Comb. and Number Theory. arXiv: math/0909.0670 Zbl1245.11008
9. J.A.M. Vermaseren, Harmonic sums, Mellin transforms and integrals. Int. J. Modern Phys. A 14 (1999), 2037–2076. arXiv: hep-ph/9806280. Zbl0939.65032MR1693541
10. J. Zhao, Wolstenholme type Theorem for multiple harmonic sums. Intl. J. of Number Theory, 4(1) (2008), 73–106. arXiv: math/0301252. Zbl1218.11005MR2387917
11. J. Zhao, Finiteness of $p$-divisible sets of multiple harmonic sums. In press: Annales des Sciences Mathématiques du Québec. arXiv: math/0303043. Zbl1295.11093
12. X. Zhou and T. Cai, A generalization of a curious congruence on harmonic sums. Proc. Amer. Math. Soc. 135 (2007), 1329-1333 Zbl1115.11006MR2276641

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