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

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 p2. If one replaces the harmonic series by ∑ k≥1 1/n for k even, then the modulus has to be changed from p2 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 Mots clefs. Multiple harmonic sums, alternating multiple harmonic sums, duality, shuffle relations. Classification math.. 11M41, 11B50, 11A07.