Mod $p$ structure of alternating and non-alternating multiple harmonic sums
Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 1, pp. 299-308.

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 $13$.

Le théorème bien connu de Wolstenholme affirme que, pour tout premier $p>3$, la $\left(p\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}1\right)$-ième somme partielle de la série harmonique est congrue à $0$ modulo ${p}^{2}$. Si on remplace la série harmonique par ${\sum }_{k\ge 1}1/{n}^{k}$ pour $k$ pair alors la congruence est vraie seulement modulo $p$ au lieu de modulo ${p}^{2}.$ On peut considérer des généralisations aux sommes harmoniques multiples (SHM) et aux aux sommes harmoniques multiples alternées (SHMA) qui sont des sommes partielles de séries zêta multiples et, respectivement, de sommes d’Euler alternées. Beaucoup de résultats dans cette direction ont été obtenus dans les articles récents [6, 7, 8, 10, 11, 12], que nous récapitulerons dans ce papier. Il apparait que, pour un premier $p$, la $\left(p\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}1\right)$-ième somme partielle d’une SHM ou SHMA générale n’est plus congrue à $0$ modulo $p$ ; cependant elle peut souvent être exprimée en terme de nombres de Bernoulli. Donc c’est un problème assez intéressant de trouver qui ils sont. Dans cet article, nous fournirons un cadre théorique dans lequel ce genre de résulats peut s’exprimer et être étudié plus longuement. Nous calculerons aussi quelques SHM supplémentaires modulo un premier $p$ quand le poids est inférieur à $13$.

DOI: 10.5802/jtnb.762
Classification: 11M41, 11B50, 11A07
Keywords: Multiple harmonic sums, alternating multiple harmonic sums, duality, shuffle relations.
Zhao, Jianqiang 1

1 Department of Mathematics Eckerd College St. Petersburg, Florida 33711, USA
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Zhao, Jianqiang. Mod $p$ structure of alternating and non-alternating multiple harmonic sums. Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 1, pp. 299-308. doi : 10.5802/jtnb.762. http://archive.numdam.org/articles/10.5802/jtnb.762/

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