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Jianqiang Zhao
Mod $p$ structure of alternating and non-alternating multiple harmonic sums
Journal de théorie des nombres de Bordeaux, 23 no. 1 (2011), p. 299-308, doi: 10.5802/jtnb.762
Article PDF | Reviews MR 2780631 | Zbl 1269.11086
Class. Math.: 11M41, 11B50, 11A07
Keywords: Multiple harmonic sums, alternating multiple harmonic sums, duality, shuffle relations.

Résumé - Abstract

The well-known Wolstenholme’s Theorem says that for every prime $p>3$ 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 $13$.

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