Theorem 2(Hoffman)
For any real, =.
Proof. We follow the same line of argument as in the preceding proof. The left-hand side is now, and a term occurs on the left-hand since once if all the are distinct, and not at all otherwise. Thus, it suffices to show (1)
To prove this, note first that the sign of is positive if the permutations of cycle-type are even, and negative if they are odd: thus, the left-hand side of (1) is the signed sum of the number of even and odd permutations in the isotropy group . But such an isotropy group has equal numbers of even and odd permutations unless it is trivial, i.e. unless the associated partition is .
Read more about this topic: Multiple Zeta Function, Symmetric Sums in Terms of The Zeta Function
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