Multiple Zeta Function - The Sum and Duality Conjectures

The Sum and Duality Conjectures

We first state the sum conjecture, which is due to C. Moen.

Sum conjecture(Hoffman). For positive integers k=n, where the sum is extended over k-tuples of positive integers with .

There remarks concerning this conjecture are in order. First, it implies . Second, in the case it says that, or using the relation between the and and Theorem 1,

This was proved by Euler's paper and has been rediscovered several times, in particular by Williams. Finally, C. Moen has proved the same conjecture for k=3 by lengthy but elementary arguments. For the duality conjecture, we first define an involution on the set of finite sequences of positive integers whose first element is greater than 1. Let be the set of strictly increasig finite sequences of positive integers, and let be the function that sends a sequence in to its sequence of partial sums. If is the set of sequences in whose last element is at most, we have two commuting involutions and on defined by and = complement of in arranged in increasing order. The our definition of is for with .

For example, We shall say the sequences and are dual to each other, and refer to a sequence fixed by as self-dual.

Duality conjecture (Hoffman). If is dual to, then .

This sum conjecture is also known as Sum Theorem, and it may be expressed as follows: the Riemann zeta value of an integer n ≥ 2 is equal to the sum of all the valid (i.e. with s₁ > 1) MZVs of the partitions of length k and weight n, with 1 ≤ k ≤n − 1. In formula:

For example with length k = 2 and weight n = 7: