integral identity on products gives Two parameters case In the particular case of only two parameters we have (with s > 1 and n, m integers): where are the generalized harmonic numbers. Multiple zeta functions are known to satisfy what is known as MZV duality, the simplest case of which is the famous identity of Euler: where Hn are the harmonic numbers. Special values of double zeta functions, with s > 0 and even, t > 1 and odd, but s+t = 2N+1 (taking if necessary ζ(0) = 0): Note that if we have irreducibles, i.e. these MZVs cannot be written as function of only. Three parameters case In the particular case of only three parameters we have (with a > 1 and n, j, i integers): Euler reflection formula The above MZVs satisfy the Euler reflection formula: for Using the shuffle relations, it is easy to prove that: for This function can be seen as a generalization of the reflection formulas. Symmetric sums in terms of the zeta function Let , and for a partition of the set , let . Also, given such a and a k-tuple of exponents, define . The relations between the and are: and Theorem 1 (Hoffman) For any real , . Proof. Assume the are all distinct. (There is no loss of generality, since we can take limits.) The left-hand side can be written as . Now thinking on the symmetric group as acting on k-tuple of positive integers. A given k-tuple has an isotropy group and an associated partition of : is the set of equivalence classes of the relation given by iff , and . Now the term occurs on the left-hand side of exactly times. It occurs on the right-hand side in those terms corresponding to partitions that are refinements of : letting denote refinement, occurs times. Thus, the conclusion will follow if for any k-tuple and associated partition . To see this, note that counts the permutations having cycle type specified by : since any elements of has a unique cycle type specified by a partition that refines , the result follows. For , the theorem says for . This is the main result of. Having . To state the analog of Theorem 1 for the , we require one bit of notation. For a partition of , let . Theorem 2 (Hoffman) For any real , . Proof.