We consider several identities
involving the multiple harmonic series
which converge when the exponents ij 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 ij 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.
