Decomposition of products of Riemann zeta values

The classical Euler decomposition theorem expresses a product of two Riemann zeta values in terms of double Euler sums. Also, the sum formula expresses a single Riemann zeta value in terms of multiple zeta values of depth n. In this paper, we express a product of n Riemann zeta values in terms of multiple zeta values of depth n. We establish a new and general explicit formulaζ(d1+2)ζ(d2+2)⋯ζ(dn+2)=∑|α|=|d|+2n−1ζ(α1,α2,…,αn+1)∑σ∈Snσd{(hndn+1)(hn−1dn−1+1)⋯(h2d2+1)}, where hj=dj+2+∑ℓ≥j(αℓ−dℓ−2)hj=dj+2+∑ℓ≥j(αℓ−dℓ−2), SnSn is the symmetric group of n   objects, and σdσd is the induced permutation of σ   on the set {d1,d2,…,dn}{d1,d2,…,dn}. The case n=2n=2 gives the Euler decomposition theorem.