Quasi-symmetric functions and mod p multiple harmonic sums

Abstract

We present a number of results about (finite) multiple harmonic sums modulo a prime, which provide interesting parallels to known results about multiple zeta values (i.e. infinite multiple harmonic series). In particular, we prove a ‘duality’ result for mod p harmonic sums similar to (but distinct from) that for multiple zeta values.We also exploit the Hopf algebra structure of the quasi-symmetric functions to performcalculations with multiple harmonic sums mod p, and obtain, for each weight n through nine, a set of generators for the space of weight-n multiple harmonic sums mod p. When combined with recent work, the results of this paper offer significant evidence that the number of quantities needed to generate the weight-n multiple harmonic sums mod p is the nth Padovan number (OEIS sequence A000931).