MULTIPLE HARMONIC SERIES

We consider several identities involving the multiple harmonic series [GRAPHICS] which converge when the exponents i(j) are at least 1 and i1 > 1. There is a simple relation of these series with products of Riemann zeta functions (the case k = 1) when all the i(j) exceed 1. There are also two plausible identities concerning these series for integer exponents, which we call the sum and duality conjectures. Both generalize identities first proved by Euler. We give a partial proof of the duality conjecture, which coincides with the sum conjecture in one family of cases. We also prove all cases of the sum and duality conjectures when the sum of the exponents is at most 6.